DIFFERENCE EQUATIONS WITH APPLICATIONS TO QUEUES
.
.
..
,
".,,*. ..
...
,.X
...... ,.".. ~
.I,
.
I
.
....-.,.X.I. ..,
I.
. ".. , , .
...I.
.
.
.
...
.. . . .-
. ......
....
._., .
, ,
I.
PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes
EXECUTIVE EDITORS Zuhair Nashed University of Delaware Newark, Delaware
Earl J. Taft Rutgers University New Brunswick, New Jersey
EDITORIAL BOARD M. S. Baouendi Universityof California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology Rutgers
Ani1 Nerode Cornel1 University Donald Passman University of Wisconsin, Madison Fred S. Roberts University
S. Kobayashi Universityof California, Berkeley and
David L. Russell Virginia Polytechnic Institute State University
Marvin Marcus Universityof California, Santa Barbara
Walter Schempp Universitat Siegen
W. S. Massey Yale University
Mark Teply Universityof Wisconsin, Milwaukee
MONOGRAPHS AND TEXTBOOKSIN PURE AND APPLIED MATHEMATICS 1. K. Yano, Integral Formulas in Riemannian Geometry (1970) 2. S. Kobayashi, Hyperbolic Manifolds and Holomorphlc Mappings (1 970) 3. V. S. Vladimimv,Equations of MathernatlcalPhysics(A.Jeffrey,ed.;A.Littlewood, trans.) (1 970) 4. B. N. Pshenichnyi,NecessaryConditionsforanExtremum(L.Neustadt,translation ed.; K. Makowski, trans.) (1971) (1971) 5. L. Nariciet al., Functional Analysis and Valuatlon Theory 6. S. S. Passman, Infinite Group Rings (1971) A Ordinary Representation Theory. 7. L.Domhoff,GroupRepresentationTheory.Part Part B: Modular Representation Theory (1971, 1972) 8. W. Boothbv and G. L. Weiss, eds.. Svmmetric %aces (1972) (1972) 9. Y. Matsushima, Differentiable Manifoids (E. T. Kobayashi, trans.) IO. L. E. Ward, Jr., Topology (1972) 11. A. Babakhanian. Cohomoloaical Methodsin GrouD Theorv _(19721 . . 12. R. Gilmer, Multiplicative Ideal Theory (1972) 13. J. Yeh, Stochastic Processes and the Wiener Integral(1973) 14. J. Barns-Neto, Introduction to the Theory of Distributlons (1973) 15. R. Larsen, Functional Analysis (1973) 16. K. Yano and S. Ishiham, Tangent and Cotangent Bundles(1 973) 17. C. Pmcesi, Rings with Polynomial Identities (1973) 18. R. Hennann, Geometry, Physics, and Systems (1973) 19. N.R. Wallach, Harmonic Analysis on Homogeneous Spaces (1973) (1973) 20. J. Dieudonnd, Introduction to the Theory of Formal Groups 21. l. Vaisman, Cohomology and Differential Forms (1973) 22. B.-Y. Chen, Geometry of Submanifolds (1973) two parts) (1973, 1975) 23. M. Marcus, Finite Dimensional Multilinear Algebra (in 24. R. Larsen, Banach Algebras (1973) 25. R. 0.Kujala and A.L.Viffer,eds.,ValueDistributionTheory: Part A; PartB: Deficit and Bezout Estimates by Wllhelm Stoll(1973) 26. K. B. Stolarsky, Algebraic Numbers and Diophantine Approximation (1974) 27. A. R. Magid, The Separable Galois Theory of Commutative Rings(1974) 28. B. R. McDonald, Finite Rings with Identity(1974) 29. J. Satake, Linear Algebra (S. Koh et al., trans.)(1975) 30. J. S. Golan, Localization of Noncommutative Rings (1975) 31, G. Klambauer, Mathematical Analysis (1 975) 32. M. K. Agoston, Algebraic Topology (1976) 33. K. R. Goodearl, Ring Theory (1976) 34. L. E. Mansfield, Linear Algebra with Geometric Applications (1976) 35. N. J. Pullman, Matrix Theory and Its Applications (1976) 36. B. R. McDonald, Geometric Algebra Over Local Rings(1 976) 37. C.W. Gmetsch, Generalized Inverses of Linear Operators(1977) 38. J. E. Kuczkowskiand J. L. Gersting, Abstract Algebra (1 977) 39. C. 0.Christenson and W. L. Voxman, Aspects of Topology (1977) 40. M. Nagafa, Field Theory (1977) 41, R. L. Long, Algebraic Number Theory (1977) 42. W. F. Pfeffef, Integrals and Measures (1977) 43. R. L. Wheeden and A. Zygmund, Measure and integral (1977) 44. J. H. Curtiss, Introduction to Functionsof a Complex Variable(1978) 45. K. Hrbacek and T. Jech, Introduction to SetTheory (1978) 46. W. S. Massey, Homology and Cohomology Theory(1 978) 47. M. Marcus, introduction to Modern Algebra (1978) 48. E. C. Young, Vector and Tensor Analysis(1 978) 49. S. B. Nadler, Jr., Hyperspaces of Sets (1978) 50. S. K. Segal, Topics in Group Kings (1978) 51. A. C.M. van RooJ, Non-Archimedean Functional Analysis(1978) 52. L. Corwin and R. Szczafba, Calculus in Vector Spaces (1979) 53. C. Sadosky, Interpolation of Operators and Singular Integrals(1979) 54. J. Cmnin, Differentlal Equations (1980) 55. C.W. Gmetsch, Elements of Applicable Functional Analysis(1980)
56. 57. 58. 59. 60. 61. 62. 63.
64.
65. 66. ... 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93.
94. 95. 96. 97 98. I
99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112.
l. Vaisman, Foundations of Three-Dimensional Euclidean Geometry(1980)
H.l. Freedan, Deterministic Mathematical Models In Population Ecology (1 980)
S, B. Chae, Lebesgue Integration(1980) C. S. Rees et el,, Theory and Applications of Fourier Analysls (1 981) L. Nachbin, Introduction to Functional Analysis (R.M. Aron, trans.)(1981) G, Omch and M. Onech, Plane Algebraic Curves (1 981) R.Johnsonbaugh and W. E. Pfafenberger,FoundationsofMathematicalAnalysis (1981) W, L. Voxman and R. H. Goetschel, Advanced Calculus (1981) L. J. Cowin and R. H.Szczafba, Multivariable Calculus 1982) V. l. lstr#escu, Introduction to Linear Operator Theory (1 81) R. D. J&vinen, Flnite and infinite Dimensional Linear Spaces(1981) J. K. Beem andf. E. Ehriich, Global Lorentzian Geometry(1981) D. L. Atmacost, The Structureof Locally Compact Abelian Groups(1981) J. W. Brewerand M. K, Smith, eds., Emmy Noether: A Tribute (1981) K. H.Kim, Boolean Matrix Theory and Applications (1982) T. W. Wieting, The Mathematical Theoryof Chromatic Plane Ornaments(1982) D. B.Gauld, Differential Topology(1982) R. L. Faber, Foundationsof Euclidean and Non-Euclidean Geometry (1983) M. Cameli, Statistical Theory and Random Matrices (1983) J. H. Canuth et al., The Theory of Topological Semigroups (1983) R. L. Faber, Differential Geometry and Relativity Theory (1983) S. Bamett, Polynomials and Linear Control Systems(1983) G. Kapilovsky, Commutative Group Algebras (1983) F. Van Oystaeyenand A. Verschoren, Relative Invariantsof Rings(1983) l. Vaisman, A First Course In Differential Geometry(1984) G.W. Swan, Applications of Optimal Control Theoryin Blomediclne (1984) T. Petrie andJ. D. Randall, Transformation Groups on Manifolds(1984) K,Goebel and S. Reich, Unlform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (1 984) T. Albu and C. Ndst&escu, Relative Finiteness in Module Theory (1984) K. HnSacek and T. Jech, Introduction to Set Theory: Second Edition (1984) F. Van Oystaeyen andA. Verschoren, Relative Invariants of Rings(1984) B. R. McDonald, Linear Algebra Over Commutative Rings(1984) M. Namba, Geometry of Projective Algebraic Curves (1984) G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics(1985) M. R. Bmmner et al., Tables of Dominant Weight Multiplicities for Representationsof Simple Lie Algebras(1 985) A. E. Fekete, Real Linear Algebra(1985) S. B. Chae, Holomorphy and Calculus in Normed Spaces(1 985) A. J. Jeni, Introduction to Integral Equations with Applications (1985) G.Kapilovsky, Projective Representationsof Finite Groups(1985) L. Narici and E. Beckenstein, Topological Vector Spaces (1 985) J. Weeks, The Shape of Space(1985) P.R. Gribik and K. 0.Kortanek, Extrema1 Methods of Operations Research(1985) &A.Chao and W. A. Woyczynski,eds.,ProbabilityTheoryandHarmonicAnalysis (1986) G. D. Crown et al., Abstract Algebra(1 986) J. H.Canuth et al., The Theoryof Topological Semigroups, Volume2 (1986) R. S. Dofan and V. A. 6el17, Characterizations of C*-Algebras (1986) M. W. Jeter, Mathematical Programming(1986) M. Altmen, AUnifiedTheory of NonlinearOperatorandEvolutionEquatlonswith Applications (1 986) A. Verschoren, Relative Invariants of Sheaves(1987) R. A. Usmani, Applied Linear Algebra(1987) P. Blass and J. Lang, Zariski Sutfaces and Differential Equations in Characteristic p z 0 (l 987) J. A. Reneke et al., Structured Hereditary Systems(1987) H.Busemann and B. B. fhadke, Spaces with Distinguished Geodesics (1987) R. Ha&, Invertibility and Singularity for Bounded Linear Operators (1988) G. S. Ladde et al., Osclllation Theory of Differential Equations with Deviating Arguments (1987) L. Dudkin et al., lteratlve Aggregation Theory(1987) T. Okubo, Differentlal Geometry(1 987)
B
113. 114. 115. 116. 117. 118. 1 19. 120. 121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151, 152. 153. 154. 155. 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170. 171.
D. L. Stancl and M. L. Stancl, Real Analysis with Point-Set Topology (1 987) T. C. Gad, Introduction to Stochastic Differential Equations (1988) S. S.Abhyankaf, Enumerative Combinatoricsof Young Tableaux (1988) H. Strade and R. Fernsteiner, Modular Lie Algebras and Their Representations (1988) J. A. Huckaba, Commutative Rings with Zero Divisors (1988) W D. Wdlis, Combinatorial Designs (1988) W Wips/aw,Topological Flelds (1 988) G. Karpilovsky, Field Theory (1 988) S.Caenepeel and F. Van Oystaeyen, Brauer Groups and the Cohomology of Graded Rings (1 989) W. Kozlowski, Modular Function Spaces (1988) E. Lowen-Colebunders, Function Classes of Cauchy Continuous Maps (1989) M. Pave/, Fundamentalsof Pattern Recognition (1989) V. Lakshmikanfham et al., Stability Analysisof Nonlinear Systems (1989) R. Sivaramakrishnan,The Classical Theoryof Arithmetic Functions (1989) N. A. Watson, Parabolic Equations on an Infinite Strip (1989) K. J. Hastings, Introduction to the Mathematlcs of Operations Research (1989) 6. Fine, Algebraic Theoryof the Bianchi Groups (1989) D. N. Dikranjan et al., Topological Groups (1989) J. C. Morgan /l, Point Set Theory (1990) P. 8ilerandA. Wtkowski, Problems in Mathematical Analysis (1990) H. J. Sussmann, Nonlinear Controllability and Optimal Control (1990) &P. Flomns et al., Elements of Bayesian Statistics (1990) N. Shell, Topological Fields and Near Valuations (1990) 8. F. Doolin and C. F. Martin, Introduction to DifferentialGeometryforEngineers (1990) S. S. Holland, Jr., Applied Analysls by the Hilbert Space Method (1 990) J. Oknlnski, Semigroup Algebras (1990) K. Zhu, Operator Theory In Function Spaces (1 990) G.6. Price, An Introduction to Multicomplex Spaces and Functions (1991) R. 6. Darst, Introduction to Linear Programming (1991) P. L. Sachdev, Nonllnear Ordinary Differential Equations and Their Applications (1991) T. Husain, OrVlogonal Schauder Bases (1991) J. Foran, Fundamentalsof Real Analysis (1991) W. C.Brown, Matrices and Vector Spaces (1991) M. M. Rao andZ. D. Ren, Theory of Oriicz Spaces (1991) J. S. Golan and T. Head, Modules and the Structuresof Rings (1991) C. Small, Arithmetic of Finite Fields (1 991) K. Yang, Complex Algebraic Geometry (1991) D. G.Hofmanetal., CodingTheory(1991) M. 0.Gonzdlez, Classical Complex Analysis (1992) M. 0.Gonzdlez, Complex Analysis (1 992) L. W. Baggett, Functional Analysis (1992) M. Sniedovich, Dynamic Programming (1992) R. P. Aganval, Difference Equations and Inequalities (1992) C. Bmzinski, Biorthogonality andIts Applications to Numerical Analysis (1992) C.Swartz, An lntroductlon to Functional Analysis (1992) S. 6. Ned& Jr., Continuum Theory (1992) M. A. ACGwaiz, Theory of Distributions (1992) E. Perry, Geometry: Axiomatic Developments with Problem Solving (1992) E. Castillo and M. R. Ruiz-Cobo, Functional Equations and Modelling in Science and Engineering (1992) A. J. Jem', IntegralandDiscreteTransformswithApplicationsandErrorAnalysis (1992) A. Chadieret al., Tensors and the Clifford Algebra (1992) P, Mer and T. Nadzieja, Problems and Examples in Differential Equations (1 992) E. Hansen, Global Optimization Uslng Interval Analysis (1992) S. Guem-DelabriBm,Classical Sequences in Banach Spaces (1992) Y. C. Wong, Introductory Theoryof Topologlcal Vector Spaces (1992) S. H. Kulkami and 6. V. Limaye, Real Function Algebras (1992) W. C. Brown, Matrices Over Commutative Rings (1 993) J. Loustau and M. Dillon, Linear Geometry with Computer Graphics (1993) W. V. Petryshyn, Approximation-Solvability of NonlinearFunctionalandDifferential Equations (1993)
E. C. Young, Vector and Tensor Analysis: Second Edition .(1993) T A. Bick, Elementary Boundary Value Problems (1993) M. Pavel, Fundamentalsof Pattern Recognition: Second Edition (1993) S. A. Albeverio eta/., Noncommutative Dlstributions (1993) W. Fulks, Complex Variables (1 993) M. M. Rao, Conditional Measures and Applications (1 993) A. Janicki and A.Wemn,SimulationandChaoticBehavior of a8table Stochastic Processes (1 994) 179. P. Neittaanmekiand D. Tiba, Optimal Control of Nonlinear Parabolic Systems (1994) 180. J. Cmnin, Differential Equations: Introduction and Qualitative Theory, Second Edition (1994) 181. S. Heikkild and V. Lakshrnikantharn, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations (1994) of Stochastic Dlfferential Equations (1994) 182. X. Mao, Exponential Stability 183. 6. S. Thornson,Symmetric Properties of Real Functions (1 994) 184. J. E. Rubio, Optimization and Nonstandard Analysis (1 994) 185. J. L. 6ueso eta/., Compatibility, Stability, and Sheaves (1995) 186. A. N. Michel and K. Wang, Qualitative Theory of Dynamical Systems (1995) 187. M. R. Darnel, Theory of Lattice-Ordered Groups (1 995) 188. 2. Naniewicz and P. D. Panagiotopoulos,MathematicalTheoryofHemivariationai Inequalities and Applications (1995) 189. L. J. Corwin and R. H. Szczarba, Calcuius in Vector Spaces: Second Edition (1995) 190. L. H. €he et al., Oscillation Theory for Functional Dlfferential Equations (1995) 191. S. Agaian et ab, Binary Polynomial Transforms and Nonlinear Digital Filters (1995) 192...M. l. Gil: Norm Estimations for Operation-Valued Functions and Applications (1995) 193. P. A. Grillet, Semigroups: An Introduction to the Structure Theory (1995) 194. S. Kichenassamy, Nonlinear Wave Equations (1 996) 195. V. F. Kmtov, Global Methods in Optimal Control Theory (1996) 196. K. l. Beidaret ab, Rings with Generalized Identities (1996) 197. V. l. Amautov et al.,Introduction to theTheoryofTopologicalRingsandModules (1 996) 198. G. Sierksrna, Linear and Integer Programming (1996) 199. R. Lasser, Introduction to Fourier Series (1996) 200. V. Sima, Algorithms for Linear-Quadratic Optimization (1996) 201. D. Redrnond, Number Theory(1 996) 202. J. K. 6eem et al., Global Lorentzian Geometry: Second Edition (1996) 203. M. Fontana et al., Prllfer Domains (1 997) 204. H. Tanabe, Functional Analytic Methods for Partial Differential Equations (1997) 205. C. Q. Zhang, Integer Flows andCycle Covers of Graphs (1997) 206. E. Spiegeland C. J. O’Donnell, Incidence Algebras (1997) 207. 6. Jakubczyk and W. Respondek, Geometry of Feedback and Optlmal Control (1 998) 208. T W Haynes et ab, Fundamentals of Domination in Graphs (1998) 209. T W Haynes et al., Domination in Graphs: Advanced Topics (1998) 210. L. A. D’Alotto et al., A Unified Signal Algebra Approach to Two-Dimensional Parallel Digital Signal Processing (1998) 211. F. Halter-Koch, Ideal Systems (1998) 212. N. K. Govil etal., Approximation Theory (1998) 21 3. R. Cmss, Multivalued Linear Operators (1998) 214. A.A.Mattynyuk, Stability by Liapunov‘s Matrix Function Method with Applications (1998) 215. A. Favini andA. Yagi, Degenerate Differential Equations in Banach Spaces (1999) 216. A.Manes and S. Nadler, Jr., Hyperspaces:FundamentalsandRecentAdvances (1999) 217. G. Kat0 and D. Struppa, Fundamentals of Algebraic Microlocal Analysis (1999) 218. G. X . Z . Yuan, KKM Theory and Applications in Nonlinear Analysis (1 999) 219. D. Motmanu and N. H. Pavel, Tangency, Flow Invariance for Differential Equations, and Optimizatlon Problems (1999) 220. K. Hrbacek and T. Jech, Introductionto Set Theory, Third Edition (1999) 221. G. E. Kolosov, Optimal Designof Control Systems (1999) 222. N. L. Johnson, Subplane CoveredNets (2000) 223. 6. Fine and G. Rosenbeger, Algebraic Generalizationsof Discrete Groups (1999) 224. M. Vdth, Volterra and Integral Equations of Vector Functions (2000) 225. S. S.Miller and P. T. Mocanu, Differential Subordinations(2000) 172. 173. 174. 175. 176. 177. 178.
226. R. Li et al.,GeneralizedDifferenceMethodsforDifferentialEquations:Numerical Analysis of Finite Volume Methods (2000) 227. H. Li and F. Van Oystaeyen, A Primer of Algebraic Geometry(2000) 228. R. P. Aganval, Difference Equations and Inequalities: Theory, Methods, and Applications, Second Edition(2000) 229. A. 6.Kharazishvili, Strange Functions in Real Analysis(2000) (2000) 230. J. M. Appell et al., Partial Integral Operators and Integro-Differential Equations inMathematicalPhysics 231. A. l. Prilepko et al., MethodsforSolvingInverseProblems (2000) 232. F. Van Oystaeyen, Algebraic Geometryfor Assodative Algebras(2000) 233. D. L. Jageman, Difference Equationswith Applications to Queues (2000) 234. D. R.Hankerson, D. G. Hoffman, D. A. Leonard, C.C. Lindner,K. T. Phelps, C. A. Rodger, J. R. Wall CodingTheoryandCryptography:TheEssentials,Second Edition, Revised and Expanded(2000) Additional Volumesin Preparation
DIFFERENCE EQUATIONS WITH APPLICATIONS TO QUEUES
David L. Jagerman RUTCOR-Rutgers Center for Operations Research Rutgers University Piscataway, New Jersey
MARCBL
MARCELDEKKER, INC. DBKKBR
NEWYORK BASEL 9
ISBN: 0-82474388-X
This book is printed on acid-free paper. Headquarters Marcel Dekker, Inc. 270 Madison Avenue, New York, NY 10016 tel: 212-696-9000; fax: 212-685-4540 Eastern Hemisphere Distribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41-61-261-8482; fax: 41-61-261-8896 World Wide Web http://www.dekker.com
The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the headquarters address above. Copyright 0 2000 Marcel Dekker, Inc. All Rights Reserved.
Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Current printing (last digit): 10987654321 PRINTED IN THE UNITED STATES OF AMERICA
To my patient and loving wife, Adrienne
Preface
The field of application of differenceequations is very wide, especially inthe modeling of phenomena, which increasingly are being considered discrete. Thus, instead of the usual formulation in terms ofdifferential equations, one encounters difference equations. In many cases a formulation using a discrete independent variable will suffice,but this severelyrestricts the available solution procedures. Further, it is often desirable to embed the original formulation into the wider class of difference equations with analytic independent variable, which not only enlarges the class of solution procedures but opens the way to answering questions concerning sensitivity, that is, to differentiation and to wider methods of approximation. The author, throughout his working career, has served in the capacity of mathematical consultant. The problems brought to him were of a varied nature, drawn from engineering, communication, physics, information theory, and astronomy; he was expected to use his mathematical knowledge to produce practically relevant and usable results. The mathematical tools that were employed included differential equations,Volterra integral equations, probability theory, and, especially,difference equations. The techniques introduced in this book were particularly useful in the construction of approximations and solutions for many of the practical problems with which he dealt. N. E. Norlund his in great work of 1924, Vorlesungen Cber Differenzenrechnung, introduced a generalization of the Riemann integral V
vi
Preface
that is the basis of this work and that I call the Norlund sum. It constructively provides a solution of the difference equation +F(z) = $(z) that reduces to a solution of the differential equation DF(z) = $(z) for W + 0. It is used to represent functions important in the difference calculus and allows representations and approximations to be obtained. It is particularly important in the solution of difference equations and various types of functional equations. Chapter 1 is a general overview of the operators and functions important in the difference calculus. Chapter 2 considers the genesis of difference equations and provides a number of examples. Also, Casorati’s determinant is introduced and Heyman’s theorem is proved. A criterion in terms of asymptotic behavior for the linear independence of solutions, due to Milne-Thomson, is given. Chapter 3 defines the Norlund sum, introducing many of its properties and, by a summability method, extending the range of its domain. Representations for the sum are obtained by means of an EulerMaclaurin expansion. The homogeneous Norlund sum is defined and an integral representation is obtained for summands that are Laplace transforms. It is shown that the homogeneous sum admits exponential eigenfunctions with explicitly defined eigenvalues.An excellent approximation for the sum in terms of the eigenvalues is derived that is also a lower bound for completely monotone functions. The value of the representations for practical computations is illustrated. This chapter is intended to introduce the reader to the properties and the use of the Norlund sum; the presentation is largely intuitive especially concerning the asymptotic properties of the Euler-Maclaurin representation, which are rigorously treated in Chapter 4. Chapter 4 presents the Norlund theory of the real variable EulerMaclaurin representation of the Norlund sum and the justification of the asymptotic relations used in Chapter 3. Fourier expansions for the Norlund sum are also studied and examples are given. An interesting class of linear transformations of analytic functions is studied using a development somewhat different from that usually presented [l], [2]. This permits the representation of difference and differential operators in a convenient form for approximationsand the solution of related equations. In particular, the Euler-Maclaurin representation for the Norlund sum is extended to the complex plane; also, an integral representation is obtained for the sum applicable to a specific class of analytic functions. In Chapter 5, a study is made of the first-order difference equation, both linear and nonlinear. The method of Truesdell [3] for differential-difference equations is discussed and applied to a queueing model, A class of functional equations of the form G($(z)) - Z(z)G(z)= m(z) is introduced and applied to the solution of a feedback queueing model. A U-operator method
Preface
vii
is constructed that is an analogue of the Lie-Grobner theory for differential equations [4].This allows the determination of approximate solutions of these functional equations.Aperturbation solution of f)Z(t)= O(z) is obtained and Haldane's method is also developed for this equation. Simultaneous first-order nonlinear equations are solved approximately. Chapter 6 studies the linear difference equation with constant coefficients and also discusses some methods for partial difference equations. The classical operational methods utilizing the E and A operatorsare used. Application is made to the probability, P(t), thatan M I M I 1 queue is empty given that it is empty initially. An asymptotic development for P(t) is obtained for large t and a practical approximation is constructed that is useful for all t . Under the assumption that the principal sum of a function has a Laplace transform, a representation is obtained for the sum by means of a contour integral [5]. Chapter 7 studies the linear difference equation with polynomial coefficients. The method of depression of order and the uses of Casorati's determinant and Heymann's theorem are illustrated. The main technique for solution, however,uses the n,p operator method of Boole and MilneThompson, which constructs solutions in terms of factorial series. Application is made to the last-come-first-served (LCFS) M I M I 1 queue with exponential reneging; in particular, the Laplace transform is obtained for the waiting time distribution. An M I M I 1 processor-sharing queue is introduced [6] exemplifying a method of singular perturbation that can be useful in a variety of queueing problems [7]. It is with pleasure I acknowledge that my friend Marcel Neuts suggested I write this book and encouraged me in the endeavor. He also recommended that I speak with Maurits Dekker of the publishing house of Marcel Dekker, Inc., regarding publication. I also wish to thank my friend Bhaskar Sengupta for reading early drafts of my material and providing suggestions anda specifically crafted problem for the text. The creation of this book took far too many years and I wish to thank the editorial staff of Marcel Dekker, Inc., for their faith and encouragement throughoutthat time. I would also like tothank my daughters, Diane, Barbara, and Laurie, for their patience and support. David L. Jagerman
Contents
Preface
...................................
iii
1
FUNCTIONS
AND
OPERATORS 1. Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Factorial function-Stirling numbers . . . . . . . . . . . . . . . . 3. Betafunction-factorialSeries ................... 4. Q Function and primitives . . . . . . . . . . . . . . . . . . . . . . 5 . Laplace and Mellin transformations . . . . . . . . . . . . . . . . . 6 . Some operational formulae . . . . . . . . . . . . . . . . . . . . . 7 . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 1 4 7 10 12 17 17
2
GENERALITIES ON DIFFERENCE EQUATIONS 1. Genesis of difference equations . . . . . . . . . . . . . . . . . . . 2. The M/M/C blockingmodel . . . . . . . . . . . . . . . . . . . . . 3. The M/M/l delaymodel . . . . . . . . . . . . . . . . . . . . . . . 4. The time homogeneous first-order model . . . . . . . . . . . . . . 5 . The Euler equation . . . . . . . . . . . . . . . . . . . . . . . . . . 6 . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
21 24 24 25 25 31 iX
...
.......
~
... . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . ..l.l......i..A.*..” .......
,.^,.
Contents
X
3 ONE
NORLUND PART SUM: 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Principal solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Some properties of the sum . . . . . . . . . . . . . . . . . . . . . 4. Summation ofseries . . . . . . . . . . . . . . . . . . . . . . . . . 5. Summation by parts . . . . . . . . . . . . . . . . . . . . . . . . . 6. Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Extension ofdefinition of sum . . . . . . . . . . . . . . . . . . . 8. Repeated summation. . . . . . . . . . . . . . . . . . . . . . . . . 9. Sum of Laplace transforms . . . . . . . . . . . . . . . . . . . . . l0 . Homogeneous form and bounds . . . . . . . . . . . . . . . . . . l1. Bernoulli’s polynomials . . . . . . . . . . . . . . . . . . . . . . . 12. Computational formulae . . . . . . . . . . . . . . . . . . . . . . . 13. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32 32 33 34 37 38 39 39 40 41 44 50 56 68
4 NORLUND SUM: PART TWO 75 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 The Euler-Maclaurin expansion. 2. Existence of the principal sum . . . . . . . . . . . . . . . . . . . . 77 3. . . . . . . . . . . . . . . . . . . . . . . 84 Trigonometric expansions 4. A class of linear transformations . . . . . . . . . . . . . . . . . . 89 5. . . . . . . . 96 Applications to expansions and functional equations 6. 7. Application to the Norlund sum . . . . . . . . . . . . . . . . . . 99 8. Bound, error estimate. and convolution form . . . . . . . . . . . 101 9. Consideration of some integral equations . . . . . . . . . . . . . 102 105 10. Bandlimited functions . . . . . . . . . . . . . . . . . . . . . . . . 105 11. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
THE FIRST-ORDER DIFFERENCE EQUATION
1. 2. 3. 4. 5. 6. 7.
109
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 The linear homogeneous equation . . . . . . . . . . . . . . . . .110 The inhomogeneous equation . . . . . . . . . . . . . . . . . . . . 114 The differential-differenceequation . . . . . . . . . . . . . . . . . 122 Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Functional equations . . . . . . . . . . . . . . . . . . . . . . . . . 132 U-operator solution of A 2 = e(Z) . . . . . . . . . . . . . . . . . 138 h
xi
Contents 8. 9. l0 . 11. 12. 13. 14.
Critical points . . . . . . . . . . . . . . . A branching process approximation . A perturbation solution of A2 = O ( 2 ) h Haldane’s method for A2 = O ( 2 ) . . . Solution of G(+(z))- Z(z)G(z) = m(z) . Simultaneous first-order equations . . Problems . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 143 . . . . . . . . . . . . . . . 145 . . . . . . . . . . . . . . . 147 . . . . . . . . . . . . . . . 149 . . . . . . . . . . . . . . . 151 . . . . . . . . . . . . . . . 154 . . . . . . . . . . . . . . 162
6 THE LINEAR EQUATION WITH CONSTANT COEFFICIENTS
1. 2. 3. 4. 5. 6.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The homogeneous equation . . . . . . . . . . . . . . . . . . . . . The inhomogeneous equation . . . . . . . . . . . . . . . . . . . . Equations reducible to constant coefficients . . . . . . . . . . Partial difference equations . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
172 172 173 179 . 188 189 198
7 LINEAR DIFFERENCE EQUATIONS WITH POLYNOMIAL COEFFICIENTS 1. 2. 3. 4. 5. 6. 7. 8
.
9. 10.
200
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 Depression of order . . . . . . . . . . . . . . . . . . . . . . . . . 201 The operators n and p . . . . . . . . . . . . . . . . . . . . . . . . 203 210 General operational solution . . . . . . . . . . . . . . . . . . . . Exceptional cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 222 The complete equation . . . . . . . . . . . . . . . . . . . . . . . . The LCFS M/M/C queue with reneging-introduction . . . . . . 226 Formulation and solution . . . . . . . . . . . . . . . . . . . . . . 228 An M/M/l processor-sharing queue . . . . . . . . . . . . . . . . 234 236 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . .
......................... . . . . . . . . . . . .
237 241
. . . . . . . .
DIFFERENCE EQUATIONS WITH APPLICATIONS TO QUEUES
Operators and Functions
1. OPERATORS The operators that are of most significancein the theory to follow are D,E , A, A. These operators are defined for functions u(x) of a complex W variable x by
The operator D is, of course, the derivative operator; E is the translation operator; A is the forward difference quotient operator; and A, which corresponds to” A for W = 1, isthe forward difference operator. Other operators of interest a& V, 6,P defined by W
vu(x)= W
W W
u(x) - u(x - W ) W
9
Chapter 1
2
S u(x) = W El. u(x) =
u(x +&l)
- u(x - $0) I
W
u(x
+h@) + u(x -
$W)
W
W
and known as the backward difference quotient, the central difference quotient, and the central mean, respectively. The corresponding operators for W = 1 are designated by V, S, p, respectively. These operators are capable of repeated application; thus
+ 1) = u(x + 2), A2u(x) = A(Au(x))= U ( X + 2) - ~ U ( X+ 1) + U ( X )
E2U(X)= E(Eu(x))= Eu(x
d2 D2U ( X ) = D(Du(x)) = -u(x).
dx2 In general, one defines E' by
E' u(x) = u(x
+r)
for all complex r. The following relation holds between the operators E and A : W
fl=l+wA
(1.5)
W
thus, E' = (1 + W A)"",
Ar= W"(P - l)'.
W
(14
W
In particular, from U(X
+ h) = (1 +
WA
)h'W~(~)
(1.7)
W
and the binomial series, the following formal expansion (Newton's formula) is obtained: U(X
+ h) =
2 (hy)o'
A'u(x). W
j=o
This expansion plays the same role in the difference calculus as the Taylor series does in the differential and integral calculus. Clearly, lim A u(x) = Du(x) W+O
(1-9)
W
so that for W -+ 0, (1.8) goes over.to the Taylor expansion of u(x A.
The differences of a function may be obtained from
+ h) about
and
3
Operators Ar = ( E - l ) r ~ ( ~ ) ;
(1.10)
thus A'u(x) =
2J
(y)(-lyu(x
+r - j ) ,
(1.11)
j=o
An important special caseoccurs when u(x) isa polynomial; accordingly, let P ( X )= aox"
+ alY" + + a,.
(1.12)
a . .
Then AP(x) = nuox""
+ -
(1.13)
so that the operator A has depressed the degree of P(x) by one unit. Thus, differences of order higher than n are all zero when applied to a polynomial of degree n; also, AnP(x)= n ! ~ .
(1.14)
It follows that Newton's expansion (1.8) is an identity when applied to a polynomial. The relation (Taylor's series) m .
(1.15) implies the corresponding operator relations E = e',
D = In E.
(1.16)
From (1.5), one has (1.17)
W
in which A, = o A , hence W 1 Du(x) = 0( A ,
~( xf )A i
+3 A i -
*
a).
(1.18)
Similarly, from (1.19) one has
Chapter 1
4
Formulae (1.18) and (1.20) are often useful for numerical differentiation. When applied to polynomials, they become identities.
2. FACTORIAL FUNCTION-STIRLING
NUMBERS
Operations of the difference calculus are facilitated by use of the factorial function defined by x(") = x(x
- 1)
' '
(x - n
+ l),
(1.21)
x@)= 1,
x(-d = (x
+ 1)
1 ' ' ' .(x
+ n)
for n 2 0 and integral. For general n, one defines x@)by [8] (1.22) in which r ( x ) is the Eulerian gamma function [8]. The salient feature of the function x@)is expressed in Ax(")= nx("-') (1-23) whose proof is (1.24)
The function x(") is related to the binomial by
(;)
==$
(1.25)
hence (1.26)
Functions
Operators and
5
Using the notation d o " for dx"at x = 0, Newton's formula provides the representation of x" in terms of factorials; thus
X" =
" 1 CXO-A~O", n > 0. j!
(1.27)
j= 1
The name Stirling numbers of tlie second kind [g] is given to the coefficients in (1.27) and symbolized by Si; hence (1.28) n j= 1
Some special values are
s:=o, n > O ; Expansion of
X"
and
S;=I,
x"+*
nPO;
s',=o,
j>n.
(1.29)
by means of (1.28) and use of
x x o = xO'tl) +j x o
(1-30)
yield the relation S,:
= S',-' +j
.
(1.31)
Using the initial conditions s0=1, 0 $=o,
j>O,
(1.32)
the numbers S', may be obtained step by step. A short table of values is given in Table 1. The inverse problem, that of expanding x@) in terms of d (1 5 j 5 n) for n > 0,is solved by use of Taylor's formula. Using the at x = 0, one has notation 0'0") for DX(") (1.33) The name Stirling numbers of the first kind is given to the coefficients in (1.33) and symbolized by $; hence (1.34) n
Chapter 1
6 Table 1: Stirling Numbers of Second Kind 1
nlj
10
2
4
3
6
5
1 3 1 7 6 1 25 15 1 31 15 90 65 63 301140 350 266127 1050 1701966 7770 3025255 4622646 6951
7
1 21
1 28
8
9
1 36
1
Some special values are
g=o,
S;=I,
n>0;
s ~ = o ,j > n .
n>O;
(1.35)
(1 -36) in (1.34) yields the relation
S',,
= &" -n SA,
(1.37)
which, together with the initial conditions
S0~ = I , $ = o ,
j>O,
(1.38)
permits step-by-step determinationof Table 2. Table 2 nlj
$, . A short table of values is given in
Stirling Numbers of First Kind 1
1 -1 2 -6 24 -225 274 -120 -7351624 -1764720 -28322 -1960 -13132 6769 13068 -5040
2 1 -3 11 -50
3
1 -6 35 -15
4
5
1 -10 85
1 175
6
7
1 -21
1
8
1
Operators and Functions
7
3. BETA FUNCTION-FACTORIAL SERIES The Eulerian beta function [8], B(x, y ) is defined by
1
1
B(x,y ) =
f"(l
- t)Y"dt,
x > 0,y > 0
(1.39)
and can be expressed in terms of the gamma function by (1 -40)
in particular, AB(x, v) = -&, y 1) in which A operates with respect to x , and
+
(1.41) (1.42)
in which j > 0 is integral. Expansions of the form
(1.43) are veryuseful in the solution of difference equations. They are called factorial series of the first kind. A Newton series of the form (1.44) is called a factorial seriesof the second kind. Both series are said to be associated. The following theorems of Landau and Norlund whose proofs may be found in Ref. 8 provide some background on the nature of associated series. It is assumed that x is nonintegral. The symbol R(x) designates the real part of x . Theorem(Landau):
Associated series converge and diverge together.
Theorem (Landau): If a factorial series converges for x = xo, then it converges in the half-plane R(x) > R(xo),and converges absolutely in the halfplane R(x) > R(x0 + 1). If the series converges absolutely for x = xo, then it converges absolutely for R(x) > R(xo)
The preceeding theorems allow the introduction of the abscissa of convergence A and the abscissa of absolute convergence p. The following theo-
8
Chapter 1
rem of Landau provides the determination of h. To obtain p, the coefficients a, are replaced by la, 1. Define a,B by
Then one has Theorem (Landau): If h I: 0, then h = a;otherwise h = B. For the condition of uniform convergence, one has the following theorems. Theorem (Norlund): If the factorial series converges at x0 then it converges uniformly for -+n+q<arg(x-xo) 0 and arbitrarily small. Theorem (Norlund): If the factorial series converges at xo, then it converges uniformly for R(x) = R(x0)
+
E
in which E > 0 and arbitrarily small. Expansion of a function into a factorial series of the first kind is unique, for assume (1.46) inwhich each series is assumed to converge in some right half-plane. Multiplying both sides by x and letting x + 00 yields a. = bo. Removing the terms corresponding to j = 0 and multiplying by x(x 1) yields aj .= bj for all j > 0. Thus, an inverse factorial series can vanish identically only If all coefficients vanish. The uniqueness theory for Newton series is not as straightforward. Consider
+
n + 00, J=O
(1.47) in which use is made of the asymptotic relation
Operators and .Functions
9 (1-48)
then
gW(
x; l ) = 0,
R(x) > 1,
(1.49)
j=O
= 1, =00,
x = 1, R(x) < 1.
Thus, for R(x) =- 1, the series provides an exampleof a nullseries, Expansion, therefore, of a functionf(x) into a Newton series may not be unique. Nonetheless, the following holds true.
Theorem: Let f(x) beexpansible into a Newton serieswithconvergence abscissa h, and let it be analytic in the half-plane R(x) > I, then the expansion is unique if l p A c 1. This may be proved by setting
c W
F(x) =
A'F(1)
(1S O )
j=O
which is the assumed expansion forf(x). Because the expansion is valid for R(x) > h (h c l), one has F(j) = f ( j ) (j 2 1) and hence F(x) =
2Ajf(l)( x 7l )
(1.51)
j=O
so the expansion is unique. Thus, the convergence abscissa of null series must be greater than one. Differences of n(x) (1.43) are readily calculated; thus (1.52) which follows from (1.41). In particular,
c W
A'n(1) = (-1)'
ai j=lj+r+ 1'
from which the Newton expansion of n(x) is immediate.
(1.53)
Chapter 1
10
4. @-FUNCTIONAND PRIMITIVES For givenf(x), a function F(x) satisfying =f
(4
(1.54) will be called a primitive or a sum off@). In order to obtain a s u m of S2(x), it is necessary to introduce another important function of the difference calculus, the psi function. From the equation
r(x + 1) = xr(x)
(1.55)
satisfied by the gamma function, one obtains by differentiation
r’(x + 1) = xr(x) + r($.
(1.56)
.Setting (1.57) one has, from (1.56) on division by 1 A+(x) =
r(x + l),
-.X
(1.58)
Thus, this identifies +(x) as playing the same role in the difference calculus as ln(x) does in the infinitesimal calculus. Thus a primitive for Q(x) may be written (1.59) Let f ( x ) be expansible in a Newton series
f(x)=
2
Ajf(l)( x
j=o
7
l);
(1.60)
then a primitive is given by (1.61) Since r’(1) = - y ( y is Euler’s constant, y = 0.57721566), onehas @(l)= -y; also, from (1.58), (-1)i-l
Aj+( 1) = -, j
jz1.
Hence one has the following elegant Newton expansion:
(1.62)
Operators and Functions
11 (1 -63)
whoseabscissa of convergenceis h = 1. This series provides a practical means of computing $(x) to moderate accuracy for 1 5 x 5 2, from which, by use of (1.58), $(x) may be computed for other values of the argument. An immediate application of the sum of a function is to the summation of series. Let A W ) =f ( 4 ,
(1.64) (1.65)
Then, because AS, =f(n
+ l),
(1.66)
one has S, = F(x)l;+' = F(n
+ 1) - F(0).
(1.67)
For example, let f ( x ) = x2; then, from the Newton expansion
x2 =
(7) +2(3
(1.68)
one has
(1.69) Thus
s n =j =k1 j 2 = n( + l
)+2(
),
(1.70)
n+l
S, = in(n +(1.71) 1)(2n + 1). As another example, consider 1 f(x)=
(1.72)
and
5io.m. w
=
1
.....................................
(1.73)
............
, _
...............................
.......................
Chapter 1
12
Since (1.74)
one has (1.75) and S = F(oo) - F(1) =
(1-76).
$ I
5. LAPLACE AND MELLIN TRANSFORMATIONS The Laplace and Mellin transformations are of particular importance in applied work. As many sources of information are available [10,1l], only certain properties of the trapformations and transforms will be cited. The Laplace transform, f ( s ) , of a function f ( t ) is defined by (1.77) for various classes of functions. The correspondence betweenf(t) andf(s) will be indicated by
f(0
(1.78) where it is always assumed that f ( t ) vanishes for negative arguments. A useful class of functions is the class L defined by "f
f C 9 7
1. f ( t ) is Riemann inte rable over ( E , T ) for arbitrary E P~f(t)l dt exists. lim S,' ~f(t)ldt= io
2.
0 and T > E .
e+O+
3. There exists so, real or complex, such that lim 1 ; e-""'ft) dt exists. 4. f ( t ) has only jump discontinuities in ( E , T).*-)OO A convergence theorem is the following:
Theorem: f ( t ) E L + (1.77) converges foz R($)> R(so)and defines a function f ( s ) analytic in that half-plane withf(oo) = 0.
Define 4(t) by Ft
(1.79)
13
Operators and Functions Then integration by parts establishes
(1.80)
Theorem: f ( t ) E L =+ (1.80) converges absolutely for R(s) > R(s0). Forf(t), g(t) E L, define h(t) by (1.81) Then h(t) E L and is called the convolution product off(t) and g(t). It is often symbolized by h(t) =f@>* An important property of the convolution product is expressed in
(1.82)
Theorem: Let the transformsf(s), &S) be convergent for the same so; then the transform, l&), of h(t) = f ( t ) * g(t) is convergent at so and i(s)
=?(is)* ,g($).
Concerning the convergence abscissa itself, one has the following results.
Theorem: If the convergence abscissa, h, satisfies h L 0 then
Theorem: Iff(t) 2 0, then the convergence abscissa, h, is a real singular point A function, N ( t ) , for which
l N ( 3 d u = 0,
t
L: 0
is called a null function. One has
Theorem:
!(S)
determinesf(t) to within a null function.
Table 3 gives a short list of operational properties (a > 0, f ( t ) = df(t)/dt). The bilateral Laplace transform
Chapter 1
14 Table 3 LaplaceTransformOperationalProperties
Function
Transform
under the change of variables f ( t ) = g(- In t )
t = e-',
goes over to the form
f(s)=
(1.83)
t""f(t) dt, 0
which is called the Mellin transform (the bar is used to indicate Mellin transform). Similarly, the unilateral Laplace transform (1-77) takes the form
1
1
f(s)=
(1.84)
t""f(t)dt.
0
The convolution product for the Mellin transform is defined by (1.85) and has the following transform property: i(8)
=f(s)&).
(1.86)
Table 4 gives a short list of operational properties (a > 0, h > 0). The Mellin transform may be applied profitably to the study of inverse factorial series (1.43). Use of (1.39) provides the formula (1.87) which suggests the introduction of the function
#(t) defined
by
Operators and Functions
15
Table 4 MellinTransformOperationalProperties
Transform
Function
(1 3 8 ) j=O
and the relation (1 -89)
The function will be called the generating function of the inverse factorial series. This shows that a(x)is a Mellin transform over (0,l) and hence equivalent to a unilateral Laplace transform. To establish the interchange employed in the transition from (1.87) to (1.89), it suffices to show that W
t"#(t) = cujr"-'(l - tr'
(1.90)
j=o
converges uniformly for 0 It I1.Let h be the abscissa of convergence of Q( x) in (1.43), assuming convergence at x = h, and let a L max(1, h + 2). Then one may set x = a - 2 in (1.43); hence, using (1.42), (1.91)
Thus, for sufficiently largej (j> n), one may choose E > 0 arbitrarily so that (1 -92)
One now obtains, for R(x) = a,
Chapter 1
16
< &tu-l(l - (1 - t))-+l < E.
(1.93)
As an example of the use of (1.89), consider
c W
n(x) =
(1.94)
dx(-J),
j=1
One has W
n(x) = a
C
(1.95)
dX(7j-l);
j=O
define (1.96) Then comparison with (1.42) provides (1-97) For another example, consider the expansion of l/x2. Because 1 -= X2
-l
1
t x - l l n t dt
(1.98)
one has (1.99) and thus (1.100) The generating function $(t) of the product of two series nl(x), n2(x) ( t by ) (1.85), whose generating functions are respectively &(t), ~ $ ~ is, = 41( 0 * 4 2 ( 0
in which one must observe that r#q(t),q52(~), 4(c) = 0 for t
(1.101) 0, t > 1.
Operators and Functions
17
6. SOME OPERATIONAL FORMULAE To conclude this brief summary of operators and functions useful in difference equation theory, certain operational results concerning the operators E and D will be presented. It is useful to think of f ( x ) in terms of a power series expansion in powers of x , then, since El = 1, (1.102) (1.103) (1.104) (1.105) Because, more generally, E [ a " u ( ~= )] O~+'U(X
+ 1) = aX(aE)u(x),
(1.106)
the shift formula follows, namely (1.107) f(E)[a"u(x)l= a'lf(aE)u(x>. The corresponding results for the operator D may be derived independently or from the preceeding results for E by using the relation (1.16). They are f ( D >1 =f ( O ) ,
(1.108)
f(D)[e""u(x)]= e"f(D
+ a)u(x).
(1,109)
Examples of these operational formulae will arise in the exercises and in later applications.
PROBLEMS 1. Solve
+
d ( x ) + 3u1(x) 2u(x) = xe+,D.
2. Solve u(x
+ 2) - 3u(x + 1) + 2u(x) = x23x.
3. Show
Chapter 1
18
4. Show (Euler's transformation) ao-al+az-.'.
1 1 = 5 a o - F A a o + ?1A' 2a o - " . . 2
5. Show
+ b) = (2 sin A" cos(ax + b) = (2 sin -)" a cos [ax + b + n "$7. a+ 2 A" sin(ax
6 . Show W
n!
,x+
l)...(x+n)=~n:;'~l(xI'>.
7. Show (Vandermonde)
(x + h)'"' =
2
(;)x("-Jw.
j=O
then show j=O
10. Show
and, hence,
j+l
Operators and Functions
c
19
W
D'f(0) =
n=r
n!
S; W"" A f ( 0 ) . W
Hint: Consider the derivatives of (1 11. Show (Stirling)
+ t)" with respect to x.
12. Show
and, hence,
Hint: Consider the differences of exfwith respect to x . 13. Show (the transformation x -+ x + m)
Hint: Replace the generating function 14. Let.
then show m
G?'@) = -
C bjB(x,j + 1) j= 1
in which j- 1
b j = C - . a, v=o j - v
+(t)
by t-"+(t)
Chapter 1
20
15. Show 1 x
1 x+l
” -+ “ . I
1 x+2
c m
I
=
2-j”B(x,j
+ l).
j=O
16. Show(Waring’sformula) 1 1 a a(a+l) +..., x - a - x x(x 1) x(x l)(x 2) ” -
+- + + +
+
17. Show
1 1 1 1 1 -... Alnx=”---”--x 2x(x+ 1) 6x(x+ l)(x+2) Hint: A lnx is the Laplace transform of (1 - e-’)/Y.
2 Generalities on Difference Equations
1. GENESIS OF DIFFERENCE EQUATIONS By the genesis of a difference equation is meant the derivation of a difference equation valid for a given family of primitives. Consider the equation F(x, u(x>,p(xN= 0, (2.1) in which p ( x ) is an arbitrary periodic function of period one, and the equation F(x+l,u(x+l),p(x))=O.
(2.2)
Elimination of p(x) from (2.1) and (2.2) yields a relation of the form G(x, u(x), u(x
+ 1)) = 0.
(2.3)
Equation (2.3) is a difference equation satisfied by every member, u(x), of the family defined in (2-1). Because only the arguments x, x 1 occur in (2.3), the equation is said to be of first order. The following are examples of this procedure. Consider the family
+
u(x> = p(x)g(x).
(2.4)
Then, from
21
Chapter 2
22
one has g(x)u(x
+ 1) - g(x + l)u(x) = 0.
Important special cases are the choices g(x) = d" and g(x) = a"T(x for which one obtains
u(x
+ 1) - au(x) = 0,
#(X
x+l + 1) - -u(x) a
+ l), (2.7)
= 0,
respectively. These correspond to the geometric distribution (1 - a)& ( x = 0, 1,2, and the Poisson distribution, +(x, a ) , definedby m)
The function @(x, a)-' satisfies (2.8) with initial value u(0) = ea. Similarly, setting g(x) = d"/r(x l), one has
+
(2.10) satisfied by @(x, a ) itself with initial value u(0) = e-". The difference equations may be used as recursions for the successive Computation of u(x) at integral points. These values, in turn, may be used to form the differences at, say, x = 0, from which a Newton expansion (1.8), with W = 1, may be constructed: thus, values of u(x) may often be readily obtained at non-integral points. Systematic exploitation of this idea occurs in Chapter 5; Another example is provided by 1 u(x) = P(X>
-x
(2.1 1)
from which follows
[
A X--
u;x,l = O
(2.12)
and, hence, u(x)u(x
+ 1) + u(x + 1) - #(X) = 0.
(2.13)
This is a special case of the general Riccati equation u(x)u(x
+ 1) + a(x)u(x + 1) + b(x)u(x) + c(x) = 0.
The Clairault difference equation is obtained on considering
(2.14)
Equations Difference
on
Generalities
23
= XP(4
(2.15)
+f@(XN
in whichf(x) is prescribed. One has (2.16)
= P(X>
and, hence, U(X)
(2.17)
= X A U ( X+) f ( A u ( x ) ) .
A two-parameter family, that is, a family in which twoarbitrary periodics pl(x),p2(x) occur, has the form F(x, u w , Pl(49 P2(X)) = 0.
(2.18)
Use of (2.19) together with (2.18) provides the relation G(u(x),u(x
+ 11, u(x + 2)) = 0
(2.20)
+
which, because of the arguments x , x + 1, x 2 is called a difference equation of second order. In general, when F = 0 contains n arbitrary periodics, p l ( x ) ,. . .p&), a difference equation of nth order is obtained. Consider the equation
44 = P1( 4 a " +P 2 ( X ) b X
(2.21)
and the additional equations (2.22) Then elimination of pl(x)a",p2(x)bX considered as unknowns provides the determinant (2.23) and, hence, the second-order difference equation u(x
+ 2) - (a + b)u(x + 1) + abu(x) = 0.
(2.24)
Illustrations of difference equations arising from model formulations are plentiful. The following are some examples.
Chapter 2
24
2. THE M / M / C BLOCKING MODEL A Poisson arrival stream of a Erlangs is offered to a fully available trunk group consisting of n independent exponential servers. Let u(n,j ) designate the probability that, at an arbitrary instant oftime with the system in equilibrium, j trunks are busy. Then the balance equation for flow into state j is (j
+ l)u(n,j + 1) - (j + a)u(n,j ) + au(n,j - 1) = 0,
u(n, 1) = au(n, 01,
2
1s j
n - 1,
u(n,j)= 1.
j=O
(2.25) The quantity B(n, a) = u(n,n), which is the probability that all trunks are busy, is called the Erlang loss function; it satisfies the following difference equation:
B(n
+ 1, a)"
n+l = -B(n,
a
a)"
+ 1,
B(0, a) = 1.
(2.26)
3. THE M/M/I DELAY MODEL A Poisson arrival stream of a Erlangs is offered to an exponential server with unit mean service rate. Let u(x, t ) designate the probability that there are x units in the system at time t if the system was empty at t = 0. One has w x , t)
- u(x + 1, t ) - (1 +a)u(x, t ) + au(x
"
at
t, - au(0, t )
"
at
+ u(1, t),
u(0,O) = 1,
- 1, t), (2.27)
x=o
Equation (2.27) provides an example of a differential-difference equation. In many forms of stochastic modeling, the generic form of equation expressing time dependence is au(x7t,
- Lu(x, t )
"
at
(2.28)
in which L is an operatorwith respect to x . Such equationsare often said,to be of Fokker-Planck or semigroup type [12].
Equations Difference
on
Generalities
4. THE TIME HOMOGENEOUS FIRST-ORDER MODEL A function Z(t; z ) with Z(0;z ) = z is required that satisfies A z(t;Z) = q q t ; z ) )
(2.29)
0
in which the function O(z) is specified. This includes the usual one-dimensional theory of branching processes [12,13]. In this role Z(t; z ) considered as a function of z corresponds to the probability generating function of the population distribution at the tth generation when t is an integer; otherwise it corresponds to continuous time branching processes. This equation is studied in Chapter 5. For further discussion of stochastic modeling, one may refer to Refs. [l2 to 141.
5. THE EULER EQUATION As an illustration outside the field of stochastic modeling, one may consider the problem of the extremization of the functional [15,16]
c n
S=
F(j, uU),VU)),
v(j) = Au(j).
(2.30)
j=O
The function F(x, U ,v) is prescribed and it is supposed that suitable boundary conditions have been specified. It is required to determine ~ ( t(0)5 t 5 n). Differentiation of S withrespect to ~ ( t yields ) the followingEuler equation: (2.31) in which A operates with respect to t. In addition, the various dynamic programming formulations [l61 provide many examples of difference equations. A homogeneous linear difference equation of order n has the form an(x)u(x
+ n) + a,-l(x)U(x + n - 1) + + ao(x)u(x) = 0. *
(2.32)
’
The solution U(X) = 0 will be excluded from consideration in what follows. It will be assumed that the coefficient functions aj(x> (0 5 j 5 n) have only essential singularities because, otherwise, multiplication of the equation by a suitable entire function will remove all poles. The following are called the singular points of the difference equation: the zeros of ao(x) and an(x - n) and the singularities of aj(x) (0 fj f n).
......,.,.., . , . . . _ , ,”.,.,.,.,,,,,..,,.......~..
~ . . I
.
.
.
.
.
.
.
..
Chapter 2
26
Given any point a, x is said to be congruent to a if x - a is an integer, otherwise incongruent. The principal interest concerning (2.32) lies in finding analytic solutions. If x is restricted to be integral, then the conditions of a solution satisfying, say, prescribed initial conditions may be relaxed. In this case, the equation may be considered to provide a solution through sequential computation and may more properly be considered a recursion. Considering the second-order equation az(x)u(x
+ 2) + a1(x)u(x + 1) + ao(x)u(x) = 0
(2.33)
to be typical, and solving for u(x), one has
If u(x) is prescribed for 0 5 R(x) 2, then, for values of x incongruent to the zeros of ao(x) and the singularities of the coefficients, u(x) maybe continued to the left. Similarly, by considering
(2.35) if x is incongruent to the zeros of az(x - 2) and the singularities of the coefficients, then u(x) may be continued to the right. Thus u(x) maybe continued throughout the plane except at points congruent to the singular points of the equation. A set of functions ul(x), , U&) satisfying (2.32)is said to form a fundamental system of solutions if there is no relation of the form
+
=0
(2.36) such that for at least one x incongruent to the singular points of (2.32), the p j ( x ) are not all simultaneously zero. The pj(x) are, as introduced earlier, periodics of period one. One then has that all solutions of (2.32) are spanned by u1(x), , U&). The following theorem of Casorati enables one to determine whether a given set of solutions constitutes a fundamental system. Pl(X)Ul(X)
'''
+P"(X)U,(X)
-
Theorem (Casorati): The necessary and sufficient condition that the set ul(x), . - . , u , ( x ) should be afundamental system of (2.32) is that the Casorati determinant u1(x) * Ul(X+
u1(x+n-
*
"U,(X)
l)*.*u,(x+ 1) l ) * * * u , ( x + n - 1)
Generalities Equations on Difference
27
should not vanish for any value of x incongruent to the singular points of (2.32). ProoJ The condition is necessary, for let U,(x) (1 Ij In) be the cofactors of the last row, then
c
Uj(X)
U@) = 0,
i= 1
(2.37)
c n
.j(X
i=l
+ n - l)Uf(X)= D(x) = 0,
in which the last equation follows by assumption. Now Ui(x cofactors of the first row, hence
c n
Ui(X)Ui(X
i= 1
+ 1) are the
+ 1) = D ( x ) = 0, (2.38)
c n
i= 1
Ui(X
+ n - l)Ui(X + 1) = 0.
Equation (2.37) determines U i ( x ) / U l ( x )(2 5 i 5 n) and (2.38) determines Ui(x l)/Ul(x l), hence
+
+
(2.39) Thus one may set
" UdX)PdX) U1 ( x ) - P1 ( x ) and consequently, from (2.37), PI(~1.1 ( X )
(2.40)
+ +Pn(x)un(X) = 0
(2.41)
*
showing that u1(x), . , un(x) does not form a fundamental system. To establish the sufficiency ofthe condition, assume u1( x ) , ., un(x)does not form a fundamental system, so that a point Q incongruent to the singular points of (2.32) and a set of periodics pl(x), - ,p&), not all zero at a,can be found for which m
e
(2.42) then one also has
Chapter 2
28
(2.43)
Thus (2.44)
D(a) = 0
and the theorem is proved. Application of Casorati's theorem to (2.24) for which a", b" are known solutions yields D(x) = axbx(b- a),
(2.45)
which, for a # b, nevervanishes;hence a", b" constitute a fundamental system. However, in contrast, for the system ax sin 2nx, b", one has D(x) = d"bX(b - a) sin 2nx
(2.46)
which vanishes for all integral values of x. Thus this does not form a fundamental system. As another example, consider the equation u(x
+ 2) - xu(x) = 0
(2.47)
for which a solution set is (2.48)
For the Casorati determinant, one has (2.49)
= (--1)~+~4&r(~). Because D(x) does not vanish at points incongruent to the singular point x = 0, the set (2.48) constitutes a fundamental system. Casorati's theorem enables the general form of the solution of (2.32) to be obtained. Thus, let ul(x), . , un(x)be a fundamental system; then, from i=O
2
+
ai(x)uj(x i) = 0,
(2.50) 1 r j 5 n,
i=O
on eliminating the coefficient functions, aj(x) (0 5 i 5 n), one has
Generalities on Difference Equations
29
I =o.
(2.51)
The minors of the elementsof the first column are not zero because ul(x), , un(x)form a fundamental system,hence periodics p(x),pl(x), , pn(x)exist for which
p(x)u(x)+P1 (4u1 (x) + ' ' * + Pn(x>ufl(X> =0 with p(x) # 0;hence one may also write 4 x 1 = P I ( X ) U ~ ( X )+
*
m
*
+pn(x>un(x>-
(2.52) (2.53)
Thus (2.53) provides the general form of solutions of the nth order, homogeneous, linear difference equation (2.32). The importance of a fundamental system is now evident. The determinant of (2.51) may be used to construct a difference equation admitting a given fundamental set of solutions. For example, given ul(x)= x , u2(x)= 2' one has
I
2xl
u(x+ 1 ) x + 1 2x+1 u(x + 2) x +x 2 2x+2
=o,
(2.54)
and hence the equation is
( x - l)u(x+ 2) - (3x - 2)u(x
+ 1 ) + 2xu(x) = 0.
(2.55)
The corresponding Casorati determinant is
D(x) = 2'(x
- 1).
(2.56)
Because the singular points are 0, 1 and D(x) does not vanish at points incongruent to 0, 1, the system x, 2' is verified to be a fundamental system. One also has, from (2.53), that all solutions of (2.55) have the form
u(x>= (2.57) P l ( X ) X +P2(x)2x. A remarkable result exists for Casorati's determinant for a given difference equation, namely that it satisfies a first-order equation. That is the assertion of Heymann's theorem. Theorem(Heymann):
Casorati's determinant, D(x), satisfies
Chapter 2
30
(2.58) Multiply the first row by al(x)/an(x),and the second row by a2(x)/an(x)up add the resulting rows to the last row. to the (n - 1)st row by an-l(x)/an(x); From (2.32), one has
hence the last row of D(x + 1) becomes (2.60) Transferring this to the firstrow of the determinant establishes the theorem. It immediately follows from Heymann’s theorem that if D(x) vanishes at a point a then it vanishes at all points congruent to a. A criterion in terms of asymptotic behavior (x + 00) for ascertaining that a given system of functions constitutes a fundamental system is contained in the following theorem. Theorem(Milne-Thomson):
If
-
in which r goes through the positive integers, then the system ul(x),. , U&) is fundamental. Proof. It is supposed all the functions uj(x) existinsome Suppose they are not fundamental; then one may write
e
half-plane.
~l(x)ul(x)+’..+~n(x)un(x)=O (2.61) in which not all pj(x) (1 5 j 5 n) are zero. Let p,($ be the last nonzero periodic; then Pl(X)Ul(X)
+
*.*
+P,(X)U,(X)
= 0.
(2.62)
Thus, on dividing by u,(x + r), (2.63)
Equations Difference
on
Generalities
31
Letting r -F bo in (2.63)and using the stated value of the limits, one obtains p,(x) = 0, which is a contradiction. When the asymptotic behavior of solutions ofdifference equations is known, this result can be usefully applied.
PROBLEMS 1 . Form the difference equations satisfied by the following families:
+
XP(X> 1 &(x) 1 2. Form the Euler equation for the minimization of u(x) = p(x)2x,u(x)
c n
S=
(u(j)2
+ w(jy>,
=
2
+
u(0) = z.
j=o
3. Show that the second-order difference equation whose solutions are
is ( x - a)(x - a
+ l)u(x + 2 ) - (2x + l ) ( x - a)u(x + 1 ) + x%(x) = 0;
alsoshow
4. Using the asymptotic criterion of Milne-Thompson,.show that the func-
tions uI(x), u2(x)of Prob. 3 form a fundamental system.
Norlund Sum: Part One
1. INTRODUCTION The basic problem to which we now turn our attentionis the solution of the equation
for primitives F(xlw) given +(x), This constitutes a generalization of the corresponding problem of the integral calculus, namely the discovery of primitives F(x) satisfying DF(x) = +(x). (3.2) Progress in the integral calculus was impeded until a constructive definition was framed providing one of the primitives of (3.2). This definitionthe Riemann integral-formed the foundation for the theory of integration. Its properties allowed a fruitful theory to be developed. Similarly,one would like a constructive definition of a particular primitive of (3.1) that would possess rich analytic properties permitting a useful theory to be developed. It should provide simple representations of important functions and have means of ready asymptotic computation and approximation. For example, certainly F(xlw) corresponding to +(x) being a polynomial should also be a polynomial; such a primitive exists, as can be seen from the Newton expansion (1.8). The unique determination of F(xlw) should rest on its value at a
32
Sum:
Norlund
Part One
33
single point ratherthana specification throughoutan interval, and one would also likelim F(xlo) to reduce to a solution of(3.2)because W+O lim Af(x) = Df(x)whenever Df(x)exists. w-*o W All these properties are provided by the formulation of Norlund [17], which will now be studied.
2. PRINCIPAL SOLUTION The definition of the principal solution of Norlund willbegiven in two stages. In order to motivate the definition, (3.1) is rewritten in the form
using (1.6). Thus W
F(xlw) = -1 -E* #(x) = -[l E" E2W
+ +
+- -
+]@#(X),
(3.4)
Formally, (3.4) is a solution of (3.1), although, without restricting +(x), the series need not converge. It was found by Norlund, however, that the desired properties were not given by (3.4) without the addition of a suitable constant. The constant chosen is
in which a is arbitrary. A firm motivation for this choice will emerge when the definition is completed in the second stage. Accordingly, one has the following definition.
Definition (Norlund Principal Solution): Let both the integral and sum converge. Then the principal solution of A F(xlo) = #(x) W
or sum of #(x) is
The notation introduced by Norlund for the principal solution is
Chapter 3
34 X
F ( x J o )= S #(z) A z W
and the operation is referred to as “summing @(z)from a to x.” The notation F(x) is used whenW = l. The quantity W is called the “span” of the sum and, unless otherwise stated, isassumed to be positive. Examples of evaluations directly from the definition are I
in which
is the generalized zeta function [18].
3. SOME PROPERTIES OF THE SUM A number of properties of the sum flow directlyfrom the definition, that is, from
(3.10) One has, of course, X
zA S #(z) A = # ( X ) . w a
(3.1 1)
W
Quite simply, one obtains from (3.10) the following relations:
(3.12) X
S #(z
+ b) A z = x+b S #(z) A z; a+b W
also, for W X
(3.13)
W
0, one maywrite XlW
(3.14)
Norlund Sum: Part One
35
in which, as is usual, Ay refers to a unit increment. Let m be a positive integer; then substitution of x vw/m (0 5 U 5 m 1) in succession for x in (3.10) and addition of the resulting equations yield
+
c
m- 1
F(x + : l a )
= mF(--). W
(3.15)
V=O
This is called the multiplication theorem of the principal solution. The origin of the name will become evident when application is made to special functions such as the Bernoulli polynomials and the gamma function. Let E > 0 be arbitrarily chosen; then, for the next result, the assumption will be made that +(x) = O ( X - ’ - ~for ) x + 00. Let n be a positive integer and let A = no;then
c m
W
c c n
+(x + j W ) = W
+(x + ] W )
+
j=o
j=O
jzn
n
=W
+(x + j W )
W
+
+(x +jo) O(W
c
(3.16) ( x +jw)-””.
Use of an integral comparison gives (3.17) j=o
j=O
uniformly for x 2 0. Thus, for fixed A, (3.18) and, hence, letting A -+ 00, (3.19) From (3.10), one now has
Of course this is what was desired becauseit provides a solution of (3.2). Also, clearly, 1 m-l 1 x+o (3.21) lim F ( x 2 Iw) = F(tlo) dt. m+m m m X u=o
c
+
--l
Chapter 3
36
The integral in (3.21) is calledthe span integral. Dividing (3.15) by m,letting m + 00, and using (3.20) and (3.21), one obtains the following theorem under the condition $(x) = O(x-"'): 0
F(tlw) dt =
$(t) dt.
(3.22)
The results of (3.20) and (3.22) already provide justification for the inclusion of the integral in (3.10). In (3.22) let x = a; then
6'"
(3.23)
F(tlw) dt = 0.
An immediate application of (3.23) is the following: Let G(xlw) be a primitive of (3.1); then the principal solution is of the form F(xlw) = G ( x ~ w+)C
(3.24)
for some constant c. Substitution into (3.23) determines c and yields the formula 1
M+O
(3.25) F(xlw) = G(xlw) If,
in which the convenient notation of a vertical bar is used to represent the computation.Thus, one may construct the principal solution given any #(t)dt from a solution primitive. This is analogous to the evaluation of lox of DF(x) = $(x). An example is given by the equation A G(xlw) = xe-",
S > 0.
(3.26)
0
Using (1.6), this may be written (3.27) and hence one has W
G ( x ~ w= ) -xe-ax EO - 1
(3.28) (3.29)
by the shift formula of (1.107). Using (lS), one now has
Norlund Sum: Part One
37
1 I - e-am(1+ w a)'x.
G(xlw) =
(3.30)
W
Expansion of (3.30) into positive powers of A yields W
(3.31) Finally, using (3.25), one obtains
It may be observed that (3.32) could also be obtained from (3.7) by differentiation with respect to S.
4. SUMMATION OF SERIES The summation of series is accomplished by the following identity, easily derived from (3.10): (3.33) An example is provided by (3.8), from which one has x+no
n-l
X
C
1
Sa Z - ~w A Za - S Zw - ~ A Z = ( x + j W ) v *
(3.34)
j=O
and hence (3.35) in particular, for x = 1,
W
= 1, (3.36)
j=1
J
in which
((U)
= ((U, 1) is the ordinary Riemann zeta function.
Chapter 3
38
5. SUMMATION BY PARTS
A[u(x)w(X)] = U(X) A W ( X ) W
W
+ + V(X
W)
A #(X)
(3.37)
0
one obtains
Now, applying (3.25) to A[u(z)v(z)] W yields the result
6 A[u(z)v(z)] = u(x)v(x) a w
+W
u(t)w(t)dt = u(z)w(z),:l
(3.39)
which, incidentally, on comparison with (3.11) shows that the OperatorSA, S do not commute. Finally, using (3.39) in (3.38) and rearranging the tefms, one obtains the formula for summation by parts.
It may be observedthat thelimit of (3.40) for W ”+ O+ becomes the usual formula for integration by parts in the infinitesimal calculus. A simple example is given by x
1
(3.41)
S zqz + 1) A z , Here, one may set 1
u(x) = -
X’
w(x) = - - *
1
(3.42)
X’
hence, on applying (3.40), (3.43)
Combining the two sums gives (3.44)
NZirlund Sum: Part One
39
6. DIFFERENTIATION The formula for differentiation of the sum followsreadily from (3.10); however, to justify the operations, the assumption is now made that 4'(x) = O(x-'-') for some E > 0. One obtains (3.45) For example, consider
One has F'(x) = 1 - SF(x)
(3.47)
and hence ~ ( x =)
1
+ ce-".
(3.48)
The constant, c, is now determined by use of (3.23); thus 1 e-SX F(x) = - - 8 l-e-'' This result may be compared with (3.7).
(3.49)
7. EXTENSION OF DEFINITION OF SUM In order to extend the range of application of the definition of the principal solution of (3.1), a summability approach will be taken. The summability factor e-Axwill be used-this is Abel summability. Accordingly, one has the second stage of the definition. Definition (summability form): For A. X
lim s 4 ( z ) A z = A+O+ W
0,one defines
5 a
Az W
whenever the indicated limit exists. It is ossible in the general theory to use other summability factors such as e-"but for the purposes of this treatment the preceeding definition suffices. A function $(x) for which the sum exists will be said to be summable. Clearly, in this extended sense, (3.1 1) is still valid. In fact, all properties established for the sum to this point remain valid including the differentia-
Chapter 3
40
tion rule (3.45), which follows from the summability procedures applied to the derivative of e-"#(x). As immediate examples of the definition, one may evaluate S + O+ in (3.7) and (3.32) to obtain, respectively, (3.50)
Clearly, repeated application of summation by parts now shows that the sum of a polynomial is a polynomial. This fact will again be brought out in the study of Bernoulli polynomials, when an explicit solution will be given. The asymptotic behavior of the sum in (3.10) for small h > 0 when applied to e-hx4(x)is closely imitated by the behavior of the corresponding integral term. Thus, even if the limits, h + 0+, do notexist individually, the limit of the difference of integral and sum can exist. This provides the basic motivation for the inclusion of the integral in the definition of principal solution.
8. REPEATED SUMMATION The definition of the repeated principal sum is Fn(xlw) = wn-l S x ((x n-l
- ' ) q 5 ( ~ A) Z . 0
(3.51)
It will now be shown that
bnFn(xIw) =
#(X)*
(3.52)
One has
(3.53)
The identity (1.26) (3.54)
used in the second term of (3.53) yields
Norlund Sum: Part One
41
(3.55)
(3.56)
(3.57)
(3.58)
9. SUM OF LAPLACE TRANSFORMS Quite often functions to besummed are, in fact, transforms, so it isof interest to obtain a representation for their sum. This representation will enable accurate numerical computation to be performed and will also permit the derivation of accurate bounds. Accordingly, let f ( z )=
LW
(3.59)
e-''f(t) dt
whichisassumed to converge absolutely for z one has the following. Representation Theorem:
0,and let x 2 a
0, then
42
Chapter 3
Proof In order to establish the representation, the extended definition of the earlier section will be used. For the construction of the sum, consider first jawe-Auf(u) du; one has e-Auf(u)du =
SW
e-"
a
du
P
e-"Y(t) dt
0
(3.60)
is justified by the absolute convergence off(z) for z > 0. IM for some constant M uniformly for z 2 U , the series (3.61) j=O
is absolutely convergent; also, because (3.62) the series is absolutely and uniformly convergent for t 2 0.One now has
(3.63)
Setting (3.64)
(3.65) and hence X
Sf (z) A z = lim o
h+O+
P 0
F(h
+ t)f(t) dt.
(3.66)
Norlund Sum: Part One
43
The function F(y) is continuous for all y
F(0) = x - a -
E
[0,00] with
4
(3.67)
W , F ( W ) = 0.
Hence lim F(A
1+0+
+ t)f(t)= F(t)f(t).
(3.68)
+
Further, for A 2 0, t 2 0, one has IF(h t)l 5 M uniformly in A, t; also, for h 2 0, t 2 E > 0, 0 < S a, one has lF(A t)eg')5 M uniformly in h, t; hence
+
( W + t)f(OI 5 Me-6fV(t)l
(3.69)
uniformly for A 2 0, t 2 0. Because this is integrable on [0, m), one now has
LW
A z = dt, F(t)f(t) W
(3.70)
which is the stated representation. The following are examples of the representation formula: (3.71) (3.72) x
lnz
S1 - A z zo= - L
W
(;'"-
-t
)
1we-x' - e-01 ( y
+ In t)
dt
(3.73)
in which y is Euler's constant. The case v = 1 of (3.71) defines the generalized $-function $(xlw), that is,
The case W = l is the ordinary $ - function,(l.57); thus, +(xll)=$(x)=S-Az=Lw(T---I e-x' dt. X 1
)
1 Z
t
1-
c t
(3.75)
The integral of (3.75) is called the Gauss representation. One may express $ ( X ~ W ) in terms of $(x) as follows. From (3.74), one has (3.76)
Chapter 3
44
obtained from the substitution z = a y . Also, (3.77)
and hence +(xlw) = In w
+ +(:)
(3.78)
I O . HOMOGENEOUS FORM AND BOUNDS The identity (3.12), namely X
X
(3.79)
indicates that one may profitably study the form X
H(xlw) = S #(z) A Z. X
(3.80)
W
This form has a number of useful properties, among which is convenienceof numerical evaluation, as will become apparent in this chapter. The difference and derivative of H(xlw) have simple forms; thus,
1X
1 X
=-SC#J(Z+W)AZ-"~C$(Z)AZ @ X
W
wx
W
(3.81)
= SX" 4 ( z + w ) - 4 ( z ) A z w
W
X
= S A +(z) A Z. x u
W
To determine H'(xlw), one may differentiate (3.82)
to obtain
Norlund Sum: Part One
45
c CO
H’(x1w) = +(x) - w
q’(x + j w )
j=O
(3.83) X
= S #’(z) A z. W
The generalization of (3.83) by use of summability is as follows:
i
~ ( x l wA), = e-Az4(z) A z;
(3.84)
W
hence, by (3.83),
Letting A + 0+, one again obtains X
H’(xlo) = S #J’(z) A Z. X
(3.86)
W
X
It is now seen that the opyators A and D commute with S .That could have been expected because S comm%es with translation, th; is, x+a
X
S ~(z)Az=S~(Z+~)AZ.
xfa
x
W
(3.87)
W
Bemuse e-” is the eigenfunction for A andD, one expects this also to hold W for S . Indeed, setting X
1 Ab)=--y
1 1 - e-7’
(3.88)
one has from (3.7),
ie+
A z = wA(wa)e-ax
(3.89)
W
so that the corresponding eigenvalue is oh(&). The representation formula of the earlier section now takes the form (3.90) x ,
showing that S f ( z ) A z is itself a Laplace transform. X
W
Chapter 3
46
To proceed further, it is important to determine properties of A@). A useful inequality for this purpose is Jensen's [191; that is, letf ( x ) be convex for x in an interval I and let p(x) I. 0 for x E I , then
(3.91)
Thus, from (3.92) follows 1 - e-y 2 ye-Y/*.
(3.93)
One has, from (3.88), (3.94) and hence, using (3.93), 1
1
A'@) 1. -?+F= 0.
(3.95)
Because h(0) = -B, )\.(m)= - 1, the monotone decreasing character of Ab) establishes (3.96) Bounds for H(xlw) maynowbe (3.90). Let If(t)I 1. M , then, since In?
W
- +(X)
W
=-
/ 0
obtained from the representation of
00
e-"wA(ot)dt 2 0,
(3.97)
one has (3.98) and from
Norlund Sum: Part One
47
(3.99)
and (3.100)
(3.101)
Equation (3.99) also implies the useful inequality 1 @(x)Ilnx--, x > 0. (3.102) 2x A further result is obtained from (3.100) on assumingf(t) p 0, thus, from (3.961, x ,
! f ( ~A) z 5 -4~ f ( x ) .
(3.103)
a,
It is clear, from (3.59), thatf(t) L 0 impliesf@)(z)is of constant sign for z > 0 and alternates with respect to r, that is,
(- l)Y@)(z)p 0,
r =0,1,2,
.
(3.104)
Such a function is called completely monotone. The Bernstein theorem [l l] implies that if f ( z ) is completely monotone for z > 0, then_f ( t ) L 0; hence the condition for (3.103) maybe restated in terms of f ( z ) , namely the requirement that f(z) be completely monotone. It will now be shown that A Q ) is convex for y E [0,m). This will permit the construction of an accurate lower bound for H(x1w). Direct calculation shows that (3.105)
It is sufficient to show that A”Q) p 0 for y p 0. Setting 1 - e-y = a so that 0 5 a 1, one has 2 (1 - .)(2 - 4 , A’IQ) = (3.106) , [- ln(1 a3 thus A”@) 2 0 is implied by
[-
-
41
2 (1 - a)(2 - a)-
(3.107)
Chapter 3
48
The corresponding power series expansions are
.
-ln(l -a>
[
] =Ea,"*
a
j=O
(3.108)
m
Since (3.109) one has (3.1 10) with (3.1 11) thus, (3.1 12) Observing that 1
2
2
(3.113)
one may write a, in the form
2 j a. - -
1
"j+32j+l-k
k+l
1 2 j 1 k+l 1 1 1+3~=~k+2,=~
C-+""-c. I= 1
(3.1 14)
One further modification of the form for aj is needed. It is observed that (3.115) generates the coefficients j
&+
1- k
(3.1 16)
Norlund Sum: Part One
49
and hence, from (3.11 l), (3.1 17) The final form for ai is now (3.118) Since the double sum of (3.1 18)is monotone increasing, in fact O(ln2j), it follows that ai reaches a maximum before decreasing to zero. This occurs at j = 5, for which aj .c 1.95417. The coefficients 2 - 2-J are monotone increasing and 2 - 2" L a . for 0 5 j 5 5 (2 - 2-5 = 1.96875); hence ai 5 2 - 2-j for all j 2 0. Thus $ 'Cy)2 0 and X C y ) is convex on y 2 0. It is now possible to prove the following theorem. LowerBoundTheorem: Let f ( z ) be completely monotone and absolutely convergent for z > 0; then
Proof. The representation of (3.90) is applicable; also, one hasf(t) 2 0 as a consequence of the complete monotonicity of f ( z ) . One may nowuse Jensen's inequality, (3.91), with p ( t ) = e-"Y(t), the convex function being, of course, wA(wt). A simple calculation shows that p = -F(x)/f(x), hence the inequality of the theorem follows. Comment: Equality occurs forf(z) = e-8z (6 L 0) which however, is no,t in the set of f ( z ) considered. If the class of Laplace-Stieltjes transforms, f(z), defined by f ( z )=
e-'' d ~ ( t )
(3.1 19)
0-
were considered [F(t)monotone increasing], then e-'" would be included. Further, the case S e 0, is, in fact, also included in the set of equality but the range of values of 6 for which the sum exists has not yet been established; this will be done in Chap. 4.
50
Chapter 3
Corollary: If the sum and integral converge then
Example 1: Since X1 @(xlw)- lnx = S - A z,
(3.120)
xzw
one has > lnx @(XI@)
+ 1 - x( 1 - e-+) W
Example 2: For f ( z ) = z-',
U >
(3.121)
*
1, one has (3.122)
Similarly, the inequality of (3.103) yields 0 ° 1 j=O (x +joy
(U -
1 1)x'2xv "
+-
W
(3.123)
11, BERNOULLI'S POLYNOMIALS The Bernoulli polynomials and numbers arose in the investigations of Jacob Bernoulli (Ars Conjectandi Basilaese, 171 3) concerning the sum lk + 2k + nk. Subsequently, they have become very useful in asymptotic investigations. They can be quite conveniently studied by means of the Norlund sum theory developed here. Detailed accounts may be found in Refs., [9and 201, The Bernoulli polynomials [17], &(x), are defined by e
+
&(x) = 1, &(x) = 6uzU-'Az, 0
U
2 1.
(3.124)
The enhanced polynomials, BV(xJw),defined by
60
B,,(xlw)= uzv-' A z
(3.125) occur frequently and also allow ready deduction of properties of &(x). The substitution z = wy in (3.125) shows that W
Norlund Sum: Part One
51
0.
(3.126)
B,(xlw) = w"B,
From (3.124) follows AB,(x) = U X ' " ~ , DB,(x) = VB,-l(x),
(3.127)
which are often taken as the defining relations for B&). An expansion for &(x h) in powers of h is obtained as follows
+
(3.128)
in which use was made of the derivative relation of (3.127). The numbers B, = Bu(0)are called the Bernoulli numbers. One now has, from (3.128), B,(x) =
2 (U)dB,,-j.
(3.129)
J
j=O
Setting h = 1 in (3.128) and using the difference relation of (3.127) provides the following formula U-
1
C (y)Bj(x) = ux'", j=O
U
2 1,
(3.130)
which may be usedas a recursion for the determination of &(x). Setting x = 0 in (3.130) yields the following recursion for the B, :
c U-
Bo = 1,
1
j=O
(;)Bj = 0,
v 2 2.
(3.131)
The first few Bernoulli polynomials and numbers are given in Tables 1 and 2.
Using (3.124) in the form (3.132)
and (3.33) provides the following solution of the original problem of Bernoulli:
52
Chapter 3
Table 1 Bernoulli Polynomials Bo(x) E 1
- -1
Bl(X) = x
2
2
Bz(x)=x - x + &(x) = x3
1 6
3 2
1 2
- -2 + - x
1 + x2 - 30 5 5 3 1 B5(x) = x5 - -x4 + - x - - x 2 3 6
E4(x) = x4 - 2x3
5 1 1 B6(X) = x6 - 3x5 f - X 4 - - x z 2 2 42 B,(x)=x 7 - - x7 6 + -7x S - - x7 3 + - x 2 2 6 6 1 4 7 2 1 &(x) = x8 - 4x7 -x6 - - x4 + - x2 - 3 3 3 3 0 9 21 3 B ~ ( =~x9) - -x* 6x7 - - x 5 2x3 - - x 2 5 10 15 3 BlO(X) = X1O - 5x9 -x8 - 7x6 5x4 - -2 2 2
+-
+
+ +
+ +
+5
(3.133) equivalently, (3.134) Replacing n by n + 1 and setting x = 0, w = 1 gives jk j= 1
= Bk+l@
+ 1) - &+I k+ 1
(3.1 35)
Applying the multiplication theorem of the principal sum to B,(xlw), that is, substituting into (3.15), one has (3.136)
Norlund Sum: Part One
53
Table 2 Bernoulli Numbers Bo = 1
5 66
B10 =
-
B12
--
69 1 2730
thus, replacing x by mx and setting o = m, (3.137)
(3.138) This is the multiplication theorem for the Bernoulli polynomials. The arguments, x, o - x, are called complementary. The following considerations introduce the relation expressed by the general complementary argument theorem to be discussed in Chap. 4. From
that is, (3.140) one obtains
Chapter 3
54
F(xl
- W ) - F(x - W1 W
(3.141)
= $(x)
on replacing W by -W and assuming that the principal sum F(xl But (3.141) implies A F(x - - W ) = $(x)
- W ) exists. (3.142)
a,
from which one has F(x - W1 - W ) = F(xlw) + p ( x )
(3.143)
in which p ( x ) is a periodic of period W . This is called the complementary argument formula. The character of $(x) determines p(x). For application of these ideas to &(x), observe that (- 1)”+’Bv+1(l- x ) satisfies A(-l)”+’Bvt1(l
-X)
= (U
+l)~”,
(3.144)
which is verified directly; hence, Butl(x) = ~ - l ) v + l ~ u + l-( lx ) + P ( X ) . (3.145) Since &(x) is a polynomial, p ( x ) can only be a constant. Thus, differentiation yields the result
B,(x) = (-1)”Bv(l
- x),
(3.146)
which is the complementary argument theorem for the Bernoulli polynomials. It is now seenthat B2,,(x)is symmetric about x = and B2H1(4)= 0. From (3.127), one has
4,
B”(1) = B,, U L 2, and, from (3.146), B, = (-l)”B,(l) = (-l)”&,
(3.147) U
2 2,
(3.148)
hence &”+l
= 0,
U
L 1.
(3.149)
Further properties of the numbers B, follow from the generating function for &(x), to be derived now. Let (3.150)
(3.151)
Norlund Sum: Part One
55
g(t, x ) = AeXfI
(3.152)
One has (3.127) Ag(t, x ) = Aexr(ef- 1) = lexr,
(3.153)
and hence (3.154)
Similarly, one derives
c
UteXt B (xlw) = v tu. ew‘ - 1 v=o V!
(3.155)
Clearly, the series of (3.155) converges for It1 < 2n/w, hence B,(xlw)/u! is 0 ( ( ~ / 2 n ) ~The ) . generating function for the Bernoulli numbers follows from (3.154), namely w
t
(3.156) u=o
The known expansion [21] t -=
t
w
t2
1 -z+2c n=l t2 4n2n2
er - 1
+
(3.157)
for the generating function can be written in the form t
”
er - 1
t
t
(3.158)
n=lu=l
hence, interchanging the order of summation, t
t l --+2 e‘ - 1 2
”
c w
u=l
2
(- 1)V+’ - 0.
(3.162)
Since ((m) = 1, (3.161) gives the asymptotic behavior of B2” for U large. In view of the recursion (3.131), the Bernoulli numbers may be assumed completely known, hence (3.161) in the form (3.163)
(3.164)
12. COMPUTATIONAL FORMULAE The material of this section is concerned with various practical expansions and with the means of numerical evaluation. These expansions are derived in a formal manner with no attention given to their range of validity. This may seem somewhat unusual but, in numerical practice, one usually cannot ascertain a priori the conditions of validity; however, pointwise error estimates will be derived in terms of derivatives. The first expansion to be considered is Norlund’s version of the EulerMaclaurin formula to be studied in Chap. 4. Let (3.165) then
(3.166)
Substituting the Taylor expansion for #(x one has
+ z) in powers of z into (3.166),
Norlund Sum: Part One
57
(3.167)
Since (3.168) one obtains (3.169) and hence, x+hw
5
4(z) A z = 0
[#(z)dz + c TBu(h)q5(’”1)(x). wv
v= 1
V.
(3.170)
The special case h = 0 is of particular importance, thus
g #(z) A z = [4(z) dz + 0
c O3
U=
1
-Bu4(u-1)(x). ow V!
(3.171)
This expansion is asymptotic for x -P CO and also for o + O+; it is not usually convergent. It provides, however, excellent approximations. It is, of course, exact when @(x)is a polynomial. Let
and define Rm(xlo)by X
S 4(z) A z = Qm(XlW> W
+ Rm(XIw)*
(3.173)
Then, by (3.171)) one may take (3.174) in which m may be increased to obtainthe first nonvanishing term. An exact representation of Rm(xlo) is given in Chap. 4. An example of (3.173) and (3.174) is
Chapter 3
58 +(z) dz - $o&(x)
+(z) A z = W
+ #jwZ+'(x) + & ( X ~ W ) , (3.175)
4
W
R z ( x 1 w ) E --+"'(x). 720
Norlund's formulation of In r(x) is In r(x)= In 6
+ 8 In zAz.
(3.176)
Application of (3.175) yields
l n r ( x ) ~ ( x - ~ ) l n x - x + I n ~ + ~x ,+ c q &(X)
G
(3.177)
1 -360x3 '
which is, of course, Stirling's formula. 0 Application of (3.171) to the evaluation of f e-"Az yields 1 h(S) = - - -
(3.178) k=l
which, in fact, converges for IS1 < 2rc. The expansion could also have been obtained from (3.88) and (3.156). The asymptotic property of (3.171) implies that, for large enough m, F ( x ~ w-) Q , ( x ~ w ) + 0 , X + 00. (3.179) The function Q m ( x l w ) , therefore, provides an increasingly good approximation to F(xlo) the larger x is. It is, however, possible to obtain an approximation when x is not large by the following device. From (3.33)' one has F ( X l W ) = F(x
+ rwlw) -
x r-l
+(x + j W ) ,
W
(3.180)
/=o
hence
+ ~ w I w ) - Qm(x+ ~ w I w ) r+ Q m ( x + ~ w I w ) - C+(x + j ~ ) ;
F ( x ~ w )= F(x
1
(3.181)
W
/=o
thus r-l
F(x~w G)Q m ( X
+ ~ w I w )- C W
j=O
in which the error is precisely
+(X
+ j ~ )
(3.182)
Norlund Sum: Part One
F(x
59
+ rwlw) - Q,(x + r ~ l ~ ) .
(3.183)
In view of (3.179), one also has r- 1
r+
00
Q,(X
+ rwlw) - WC 4(x + j w ) j=O
1
(3.184)
or, equivalently, Do
F(xlw) = Q,(xl@)
+ W C [A Q& j=O
W
+j4~) -
+j~)l-
(3.185)
As an illustration of (3.182), the computation of In r(x) to an accuracy better than lo-' uniformly for x 2 1 may be accomplished by choosing Qz(x)=(x-h)lnx-x+ln&+&
(3.186)
and approximating by (3.187) Define (3.188) which is the expression used in the lower bound theorem; then L(x10) provides an excellent approximation even when summing a function that is not completely monotone or even a Laplace transform. Let R(xlw) designate the error, so that one has X
S 4(z)A z = L(xlw) W
+ R(xlw).
(3.189)
Then an estimate for R(xlo) may be obtained using (3.171) and (3.178), and the following expansion is obtained, L(XlW)
1 720 2
= --@(x)
1 + -u2#'(x) 12
1 W4 4'W3 -4(x)' +***. --
(3.190)
Use of (3.171) yields the expansion 1 1 6 4(z)A720 z = - -@(x) + 2 12
W24'(X)
X
hence
w
1 - -W44"'(x)
+-
,
(3.191)
Chapter 3
60
(3.192) An interesting implication of (3.192) is the inequality (3.193) valid for completely monotone functions with equality occurring for 4 ( x ) = e-”. Generally, however, the sign of R ( x l o ) depends on x and W . The numerical computation of Qz(xlw>, &(xlw), L(xlw), and R(xlw) may be accurately and conveniently accomplished by use of numerical differentiation. Let ah = Eh12 - E-h/2 (3.194) then the following formulae are suggested: (3.195)
1
f$”’(x) E g S&X).
The integral in the expression for Q,(x~w) maybe evaluated byuseof standard quadrature rules such as the Simpson or Gauss-Legendre rules [22]. The Gauss-Legendre rules are particularly efficient when applied to sufficiently smooth functions; an n-point rule has degree of precision 2n - 1. Writing the rule in the form (3.196) then the nodal points, z t ) , and weights, A t ) , are symmetric with respect to z = 0; that is,
At)-
- An-k+l zt’ = -zn-k+l(n). (3.197) Thus, it is sufficient to tabulate only for 0 5 z t ) 5 1. Table 3 lists the values for n = 10. The numerical evaluation of S:f(z) A z directly from (3.90) is readily effected by use of the Gauss-Laguerre qfiadrature rule. This rule takes the form 9
(3.198) and also has degree of precision 2n - 1. Writing (3.90) in the form
Norlund Sum: Part One
61
Table 3 Data for Gauss-Legendre Rule
,97390 65285 ,86506 33667 ,67940 95683 ,43339 53941 ,14887 43390
,06667 13443 .l4945 13492 ,21908 63625 ,26926 67193 ,29552 42247
(3.199) one may identify (3.199) with (3.198). Table 4 provides a list of tfo) and
AfO)
The negative numbers in parentheses indicate the power of 10 by which the A:') are to be multiplied. A quadrature evaluation proceeding bysuccessive derivatives, such as (3.170), is said to be of Euler's type. One may also obtain evaluations in terms of successive differences, in which case they are called Laplace's type. An expansion analogous to (3.170) will now be derived. From (1.8), one has (3.200) hence, using (3.166), Table 4 Data for Gauss-Laguerre Rule
,13779 34705 ,72945 45495 1.80834 29017 3.40143 36979 5.55249 61401 8.33015 27468 11.84378 58379 16.27925 78313 21.99658 58120 29.92069 70123
,30844 11158 ,40111 99292 .21806 82876 ,62087 45610 (-1) .95015 16975 (-2) .75300 83886 (-3) .28259 23350 (-4) ,42493 13985 (-6) .l8395 64824 (-8) .99118 27220 (-12)
Chapter 3
62 x+ho
5
#(z) A z = W
1’
c (‘r)
hw
+ $j
W
#(z) dz
a
(3.201)
A:#@) A z W
u=o
(3.202) The evaluation of the inner summation is
(3.203)
The numbers L , defined by (3.204)
Lu=I’(;)dV are called the Laplace numbers. One now has x+ho
5
#(z) A W z =
1’
#(z)dz
a
+
W
2 (S>
A;-’#(x)
U=
-W
1
c W
L,A:”’#(x),
u=l
(3.205) which is analogous to (3.170). The important case h = 0 is
4 #(z)A z = L x # ( z )dz W
c 00
W
L,A:-’#(x),
(3.206)
u=l
and, when a = x, (3.207) The generating function, g(t), for the L,,, namely (3.208) is readily obtained from (3.204); thus
Norlund Sum: Part One
63
s(t>= v=o l
=
-
l
(1
+ t)"dv
(3.209)
t
ln(1 + t ) '
"
whichconvergesfor Itl 1 L, = 5 L"-1
1. Thecorrespondingrecurrencerelationis
- -31 L"-2 + ;1r LP" -
- +-(-l)"+' LO v+ l
(3.210)
Table 5 lists the first few numbers. The Laplace numbersmay be written in terms of Stirling numbers of first kind; thus from (1.34),
(;)=$kSi.', j=1 hence, using (3.204), Table 5 LaplaceNumbers Lo = 1
1 L, = 2 L2= - - 1 12 1 L3 =24 L4 =-- 19 720
L6 =-- 863
60480 275 L, = 24 192 33953 L8 = -~ 3628800 5728 1 Lg =7257000 3250433 L - -479001600
''
,
,
.,
,
,
,
.
,
./
,
.. . _, ,
.,
.
. .. ...... ,. ,^.., .
,
.
. , , ...
. ,. .
,..,. .
.
I
...I_
"..l
........, , ,I..
..l/..
. , . ._.
. . .
..
Chapter 3
64
(3.212) The mean value theorem applied to (3.204) yields
L,=
G), o < e
0) cc j=O
1
1 1’
10. Obtain the following asymptotic formulae: x 1 x ” , 00, SAZ tan” x - -2x2+ oz2+1
-
Chapter 3
70
11. Let f ( z ) be completely monotone for z 2 0 and (IY L 0 and then show
12. Show 00
A(S) = - x(--l)”L,(l
- e-Sy-1,
v= 1
13, Show
14, Show that the coefficients, C,(xlw), in (3.243) can be expressed in the form
15, Define g(t) by
show 1
1 Ssms
&) = -- S-
1’
16. Define the Binet function, ~ ( x )by ,
lnr(x)=(x-{)lnx-x+In~+W(x) and 4 0 ) by q5(x)=(x+{)AInxand then show
1,
Norlund Sum: Part One
l
X
4
0
71
P ( X ) = - - S #(z)Az,
17. Obtain the expansion (S L 0)
18. It is expected that L(xlw) (3.188) will constitute an exceptionally good approximation for functions #(x) satisfying
Show that
+ +
K[cosh(cx K[sinh(cx d)I2l3, K, c, d constants, are solutions.
19 Obtain the following expansion for the Riemann zeta function: m
20.Show 1 m 0.
(4.25)
Integration by parts will be applied to (4.24); accordingly, introducing the function
Chapter 4
80
q 1 ( z ,h) =
LW
e-*W'&(h
- t ) dt,
(4.26)
one obtains (4.27) The behavior of the function q1(z, h) for h + O+ is obtained as follows:
ql(z,1.1 = e-hor
e-hwt&(h - z - t ) dt
(4.28)
Integration by parts applied to the last integral in (4.28) yields
(4.29) and, hence, (4.30) It now follows that there exists a constant c for which l+l(z,
95 Ce-*wr
(4.31)
for all h > 0 and arbitrary z. The first term of I, clearly vanishes for h + O+ because lim g1(O)exists *+O+ while the limit of the second term is zero in view of (4.15) and (4.31), hence (4.25) has been proved. For the integral Io =
LW
+
e-*(x+Wz)j m (h - z)~(")(x wz) dz,
integration by parts will again be used. Set
(4.32)
Norlund Sum: Part Two
81
(4.33) then
Because of condition (2), the limit of the second term is zero, hence
- z)+(")(x + wz) dz.
lim Io = phZ(0) =
I+O+
(4.35)
One now finally has
+$Lw
&(h - z)r$")(x
+ wz) dz,
0 5 h 5 1,
(4.36) Since all limitsare uniform for b 5 x 5 B, F(x+ hwlw) is continuous for x L: b. The expansion (3.170) is now made precise by (4.36), and the existence of the principal sum has been established under conditions (l), (2) or (l), (3). The differentiation formula of (3.45) maybederived from (4.36). Assuming conditions (l), (3), F(x + hwlw) may be written in the form F(x
+ hwlw) = [4(z) dz +
c
m 0"
v= 1
+ $1'&(h
-Bu(h)&'")(x) U!
c W
- Z)
#(")(X
+ wz + j w ) dz.
(4.37)
j=O
For m > 1, differentiation with respect to h yields
W
(m- l)!
j=O
(4.38) in whichD indicates differentiation with respectto x. Letting h + 0 in (4.38) now gives
Chapter 4
82
(4.39) Continuing the differentiation of (4.38) with respect to h up to order m, observing the jump at z = h of &(h - z), and letting h -+ 0 yield
X+'"'(x W
-W
D"F(x1w) = r$'""'(X)
+jo)
j=O
(4.40)
Hence
(4.41) and, finally,
(4.42) Thus the rincipal sum has continuous derivatives up to order m;further, since c$"- )(m) exists, one has, from (4.42),
P
lim D " F ( ~ I W=) 4(m-1)(m),
x")
(4.43)
W
This property is characteristic of the principal solution of (3.1); any other solution differs from the principal solution by a periodic, p(x), hence lim,;,p(")(x) must be a constant that can only be zero. Thus, only the principal solution has the property expressed in (4.43). The asymptotic properties of F(xlw) with respect to x + 00 and W + O+ asserted after (3.171) will now be established. As in (3.173), set
(4.44)
LW
R,(xlw) = m!
+
&,(-Z)~(")(X wz) dz;
(4.45)
then F ( x l ~= ) Qm(XIW)
+ Rm(xla),
which is (4.36) for h = 0. Conditions (1) and (2) imply
(4.46)
Norlund Sum: Part Two
83
lim R,(xlw) = 0,
x-+
(4.47)
W
hence lim [F(xlw)- Qm(xlw)]= 0,
(4.48)
X+oQ
which is (3.170). To study the asymptotic character with respectto W , the condition is now imposed that
(4.49) be convergent. This clearlyimplies condition (2). From (4.45),for some constant c,
+
lt$(m)(x wz)l dz 5 cum
IRm(xlw)l 5 cwm+* 0
ioQ
hence Iw""R,(xlw)l is uniformly bounded in x and W"
R,-l(xlw) = - B
m!
m
+ Rm(XI
(X)
lqj(m)(z)l dz,
W.
(4.50)
Also, from
W )9
(4.51)
one has
(4.52) thus
(4.53)
m 2 1.
R,(xlw) = o(w"),
Defining Ro(xlw) = - ho$(x)
+ Rl(xlw),
(4.54)
one has
(4.55) and, hence, lim F(xlw) = W+O+
la
$(z)dz,
(4.56)
which reestablishes (3.20)under condition (4.49). . It is possible to obtain a very simple bound for R,(xlw) in (4.46)that is often used in practice. For this purpose the following assumptions are made:
c#J(")(x)is continuous for x 2 b. W
C qb(*")(x + j w ) j=o
is uniformly convergent for b 5 x 5 B ( B arbitrary).
Chapter 4
a4 lim 4 ( 2 m - 1 )= ( ~0.)
x")
W
~$(~")(x) does not change sign ( x 2 b).
Observing that j h ( - z ) = &(z),
one may write m-l
F(xlw) = [ 4 ( z ) d z - ;4(x) w2m+l
S
w2u
+ ~ -1 B(2u)! 2 , 4 ( 2 u - 1 ) ( ~+ ) R2,,,,
(4.57)
v=
W
Rzrn= - ( j 2 , ( z ) - B2m)4(2")(x+ wz) dz. ( W ! 0
(4.58)
Thus:
Setting m = 2 in the multiplication theorem (3.138) yields B"(L2) = (2lnU- l)&;
(4.60)
hence, since the maximum of IB2,(z) - BzmI on (0,l) is IBzm($)- BzrnI = 2(1 - 2-2m)1B2m1,the Darboux mean value theorem applied to (4.59) yields (4.61)
Thus, the error is smaller in absolute value than twice the next term of the summation in (4.58);this is characteristic of asymptotic expansions.
4. TRIGONOMETRIC EXPANSIONS Trigonometric expansions for F(xlw) can be readily obtained whose coefficients are simply expressedin terms of 4(x). These representations are sometimes useful for direct numerical computation and usually provide ready means of truncation error estimation. The condition expressed in (4.49) willbe assumed so that F(xlo) willpossess a continuous first derivative. A Fourier series for F(xlw) will be constructed for x. < x < x,, + W , x . 2 b, of period w which, by the condition assumed, will be convergent [23]. Accordingly, set (4.62)
Norlund Sum: Part Two
85
for which the Euler formulae for the coefficients are (4.63)
2rcnx F(xlw)cosdx,
n 2 1,
(4.64)
2rcnx F(x1w) sindx,
n 2 1.
(4.65)
W
W
The integral of (4.63) is recognized to be the span integral, (3.22), hence
l
X0
a0 = 2
(4.66)
4(x) dx.
For the integrals of (4.64) and (4.65), it will be convenient to combine them into a single integral, namely 2 xo+o a, + ib, = F(xlw)e12nnx/o dx , nzl, (4.67)
--lo
in which i is the imaginary unit,
6
Consider again the function
~ ( x l oA;) = e-AZ#(z> A z ,
(4.68)
0
which is given by the uniformly convergent expansion (4.69) From (4.62), since F ( x ( o ; h + ) F(xJw)uniformly, one has
2S " ' F(xIW)ei2nnx/w " dx = lim 2 h+O+ W
F(x1o;h)ei2nnx/wdx.
(4.70)
x0
The representation (4.69) may now besubstituted into (4.70); interchange of summation and integration is permissible by uniform convergence, hence one gets
The desired coefficients are now given by (4.72)
Chapter 4
86
b, = - lim 2 A+O+
1
2nnx e-Axq5(x) sin dx. 0
(4.73)
The integrals of (4.72) and (4.73) are not usually convergent at h = 0; however, integration by parts will permit the limit to be evaluated in terms of convergent integrals. As an example, consider X
S 22A2 = & ( X )
F(x) for 0
(4.74)
0
x c 1; then (4.75)
a0 = 0,
eqhx4x cos 2nnx dx, (4.76) e-hX4xsin 2nnx dx.
In this case the integrals are easily evaluated and one finds 4 2 n 2 - h2 1 h+O+ (4x22132+ - nZn2 ' 4nnh b, = - lim 4 (4n2n2 ~ 2 =) a,
= lim 4
"
+
(4.77)
9'
hence (4.78) As another example, consider
(4.79) for 0 .c x < 1, then a0 = 0,
a, =
hence,
- A+O+ lirn
2
LW
cos 2nnxdx,
(4.80)
Norlund Sum: Part Two
87
26 4nznz ' 4nn b, = 4 9 n 2 + d2 ' a,
=-
+
(4.8 1)
and
c W
F(x) = -
x
n=l
+
26 cos 2nnx 4nn sin 2nnx 4n2n2+ sZ
(4.8 1b)
The Fourier expansion of the $-function will now be obtained for x. x0 1. Since
+
X 1
$(X)
= S - Az,
(4.82)
1z
one has a. = 2 r ' d z z a,
= - lim
= 21nxo,
w
A+O+Lo
b, = - lim A+O+
2
e-AX- cos 2nnxdx, X
(4.83)
jX0e-AX6 sin 2nnxdx. X
Using the cosine and sine integrals defined by [24] (4.84)
the formulae for a,, b, become
= 2ci(2nnxo), b, = 2si(2nnxo); a,
(4.85)
thus the expansion for $(x) is $(x) = In x.
+2
c W
(ci(2nnxo)cos 2nnx + si(2nnxo)sin 2nnx). (4.86)
n=l
When the function F(xlw) has jump discontinuities, the Fourier coefficients are O ( l / n ) and the series isnot rapidly convergent. To improve its use for numerical computation, the rate of convergence should be increased; this can be accomplishedby removing the discontinuity. For this purpose define G(xl4 by
Chapter 4
88 X
G(xlw)
(4.87)
s(+(z) - ~ ( X O )A W ) 2.
Then A G(xo(o)= 0, so no jump is present, and W
+
~ ( x l w=) ~ ( x l w ) +(xo)(x - a -
;).
(4.88)
The coefficients of G(xlw) are designated by ab, a;, b; and are given by
(4.89) e-*X(+(x)- +(xo))sin 2nnx dx. The evaluation in terms of a0,an,bn is
ab = a. - 2+(x0)(x - a
-;), (4.90)
As an example, consider the function defined in (4.79) for which a = 0, x. = 0, +(xo) = 1 ; one has
ab = 0, (4.9 1 )
the coefficients are now O ( l / n 2 ) so that the convergence has beenmuch improved. The Fourier series for G(xlw) is G(x1w) =
c
--4n2
+
s2
n= 1
cos 2znx
+
s2 nn(4n2n2+ s2)
and
+X -4
F ( x ~ u )= G ( x ~ w )
I
(4.93)
Another representation of F(xlo) will now be obtained from the Fourier series of (4.62) that is applicable for any x 2 b. If the formulae for the
Norlund Sum: Part Two
89
coefficients, (4.66), (4.72), and (4.73),are substituted into (4.62),the following equation is obtained:
(4.94) It is desired to set x = xo;however, at a point of discontinuity, the sum of the Fourier series equals the average of the left- and right-hand limits. Let S(x) denote the Fourier series for F(xlw), i.e., the right-hand side of (4.94), then
(4.95)
thus, F(xol4 = S ( X 0 )
- +#4xo).
(4.96)
Now, setting x = xo, the required expansion is
c / W
#(z) dz - i&(x) - 2
F(X~W =) a
lim
n= 1 A+O+
W
2mz + z)cos dz
0
W
(4.97) In this form, the infinite series is seen to express the difference F(xlw) and the asymptotic approximation provided by the first two terms.
5. A CLASS OF LINEAR TRANSFORMATIONS A certain class of transformations will be studied whose properties readily enable one to discuss the convergence of finite difference expansions and to solve various forms of functional equations. Application will be made later in the chapter to the extension of the Norlund expansion of (4.36)into the complex plane. Norlund's own discussion of the extension of the Norlund sum into the complex plane is in Ref. 17.A deeper version of the material to be presented here may be found in Ref. 1. A sequence of numbers ( a k ) ris of exponential order /.L if there exists some r > 0 for which
90
Chapter 4
and p = inf r.
(4.99)
The linear vector space n ( p )is the space of all functions #(z) of the complex variable z = x iy ( x , y real) given by
+
(4.100) in which the sequence is of exponential order not exceeding p. Let M(r) be the maximum modulus of #(z), that is, max)#(z)I = M(r); Izl=r
(4.101)
then, clearly, 4(z) is entire and ~ ( r=) O(e(N+’)‘),
E
> 0.
(4.102)
An entire function satisfying (4.102)is said to beof exponential type p; conversely, a function of exponential type t Ip is in n(p). This follows from Cauchy’s inequality l#(k)(0)l5 k!M(r)r-k,
r > 0.
(4.103)
+ E ) and the use of Stirling’s formula for k! show that I#(%)l = O((P + clk). (4.104) The function @(()of the complex variable ( = 6 + iq ( 6 ’ 9 real) defined by
The choice r = R / ( p
(4.105) for [(l = p > p is called the associated function of #(z). One also refers to @(()as the Bore1 transform of #(z). One has the following relation connecting #(z) and @((). Representation Theorem:
The path
is a circle of radius p > p about the origin in the (-plane.
Proof. From (4.98), there is a K > 0 for which lakl < K ( p
+
E)k,
E
0,
(4.106)
Norlund Sum: Part Two
91
hence E,
19)
(4.1
hence
and, therefore,
B, = O((P + Elr>. Also, from (4.1
(4.121)
17) and (4.118), one has (4.122)
thus O(z) E !&).
Uniqueness Theorem: The generating function L(e
(5.9)
The multiplication formula, with W = 1, applied to the principal sum W In (xlw) - w ln yields (5.10) Since, from (5.8), = In r(mx) - (mx - 1) In m,
(5.1 1)
Chapter 5
112
one has
which is the Gauss multiplication formula. The importantspecial case m = 2 is called the Legendre duplication formula, namely (5.13) The case in which a(x) has the form (5.14) can be solved in terms of the gamma function. One has 1 X
In u(x) =- S In a(z) A z, 0 0
W
-1SxC l n ( z + u j ) A z W
0 O j=1
(5.15)
1 x
- -Sxh(z+bj)Az W
W O j=1 1 X
+-SlncAz; WO
W
hence, a solution is u(x) = c"/"
r(x + al I W )
r(x+ bl IW) .
r(X
r(X
+ aUlw> +~ , I w )
(5.16)
or, equivalently, by use of (5.9), (5.17)
As a simple example, consider
+
( 2 +~ ~ ) u ( x One has
W)
+
- ( 6 ~ ~ ) u ( x= ) 0.
(5.18)
The First-Order Difference Equation
113
1
u(x+W)-3- x
+
x+l
u(x) = 0,
(5.19)
hence a solution is (5.20)
Another form of solution may be obtained by setting b = 00 in (5.3), thus
2 In a(z) 4 z X
u(x) = e
1/ U
'
n
(5.21)
W
=
a(x + j w p
j=O
Of course, convergence of the infinite product is assumed. An example is given by u(x
+ - (1 + e+>u(x) W)
=0
(5.22)
for which (5.23)
In this regard it is useful to have some criteria for the convergence of an infinite product (1 + vi). Such criteria are [25]: j
1. 0 5 vj 5 1 or -1 5 vj 5 0,
c
cjvj converges + n(l+ vj) converges J
+ n(l+ uj) diverges;
vj diverges
j
2.
j
cjuj converges + nj(l
+ uj> converges if C vj converges, i
diverges to
00
c c
if
vj diverges to 00,
j
diverges to 0 if
vj diverges to
j
3.
C uj converges absolutely =+ j
- 00,
Chapter 5
114
n(+ 1
uj) converges absolutely.
j
With respect to uniform convergence, the following theorem is useful:
Theorem: Let u j b ) be continuous for y E [a, b] for each j , and let 1 Vjb)I 1. vj 0,E [U, b]) with C uj convergent, then n ( l Uj b)) is a continuous
+
function of y for
i y E [a,
i
4.
Application of this theorem to the solution in (5.23) shows u(x) to be a continuous function for all x . The form of solution in (5.21) satisfies the boundary condition u(00) = 1 under uniform convergence. An approximation to this solution maybe obtained by use of the Norlund expansion (3.171); thus, 1 x
In u(x) = - S In a(z) A z WW
0
1X =-Sha(z)Az-WO
W O
W
In a(z) dz
(5.24)
1 W a'(x) lna(z)dz--lna(x)+--+.--. 2 12 a(x) An approximation to u(x) is, accordingly, u(x) 2: a(x)"/* exp
[-1 W
W
In a(z) dz ,
1
a'(x) +12 a(x) ' W
(5.25)
in which the error may be estimated from the next term of the expansion in (5.24),namely -(w3/720)D3lna(x). Since, for convergence of (5.21), one must have a ( m ) = 1, if also, a'(00) = 0, then the approximation satisfies the required boundary condition at x = 00. For the example of (5.22), one has u(x) N
1
4 -
(5.26)
The error estimate to be used in the exponent of (5.26) is (w3/720) ex(ex l)/(ex + l)3.It may be observed that the approximation (5.25) is particularly useful when W is small because then the product (5.21) is slowly convergent.
The First-Order Difference Equation
115
3. THE INHOMOGENEOUS EQUATION The complete equation of first order is U(X
+
W)
- U ( X ) U ( X ) = b(x),
W
> 0.
(5.27)
The solution may be constructed from the solution, u(x), of the corresponding homogeneous equation u(x
+ - a(x)u(x)= 0 ,
(5.28)
W)
which, from (5.3), may be taken to be (5.29)
The summation without a lower limit expressed is here usedas an indefinite summation symbol. Thus, let u(x) = u(x)t(x);
(5.30)
then substitution into (5.27), yields A t(x) = W
b(x) WU(X
+
(5.31)
W)*
One now has
[
u(x) = u(x) p ( x )
b(z) + -1 S A z] u(z + X
W
W)
(5.32)
W
As in the case of the homogeneous equation, the principal solution is obtained by replacing the periodic, p(x), by a constant. For many applications, a useful form of solution is l X b(z) h z u(x) = - u(x)S wu(z+ W )
W
W
(5.33)
W
+
b(x j w ) a(x +jo)*
- j=o C a(x)a(x+ w ) . -
(5.34)
The ratio test shows that the series is absolutely convergent if lim sup j+ w
+ + +
b(x jo W ) b(x j w ) a(x j w
+ +
(5.35) W)
and also uniformly convergent if (5.35) holds uniformly in x . Let lim
x-+w
a(x)a(x
+
b(x j w ) = olj, W ) ' ' * a(x + j W )
+
j 20
(5.36)
Chapter 5
116
and let (5.34) converge uniformly in x , then u(x) satisfies the boundary condition m ..
u(o0) = -
C./.
(5.37)
j=O
The general solution may now be written
Ca(x)a(x +b(x +jo)a(x + j w ) 00
u(x) = cu(x) -
W)
j=O
(5.38)
e
in which c is to be determined from the boundary condition on u(x). Returning to (5.32), the general solution may be written (5.39)
and, in particular, if u(0) is specified, (5.40)
This solution, of course, is not subject to the condition of (5.35). Approximations may be constructed to this form of solution by means of Norlund’s expansion (3.171), (3.188), or (3.206). Consider the example U(X
+
W)
- U U ( X ) = b(x)
(5.41)
in which a is constant. Since u(x) = aXjw
(5.42)
one has, from (5.39), u(x) = tax/@
+ -1 XS a(x-z)/wb(z)A z . W
(5.43)
0
Let
14,
(5.44)
then one has (5.34) (5.45)
If the series is uniformly convergent in x and la1
=- 1, b(o0) = b, then
Equation Difference First-Order The
117 b
u(o0) = -a - 1'
(5.46)
If (5.44) is met, the complete solution is
c m
u(x) = c d " -
a-j-'b(x
+j o ) .
(5.47)
j=O
The solution given by (5.40) takes the form (5.48)
For the choice b(x) = x , one has, by use of (3.32) with a = esW,
(5.49)
hence
[
u(x) = u(0)
"I
X +ax/@- -(a a - 1 (a-
W
-I
1)2
(5.50)
The case a = 1 leads to u(x) = u(0)
x'-WX +-
(5.51)
*
2W
Consider the equation U(X
+ - X U ( X ) = b(x).
(5.52)
W)
In this case, one may take (5.53)
hence (5.54)
Let
I
1 lim sup b ( x + j w + w ) j+ 00 b ( x + j w )x + j w + w
then
I
0, E ( 0 ) = E o .
(5.257)
For r = 1, one has Eft
+ h) = (1 + hBI(tx))E(t),
(5.258)
hence, a is attractive if 11
+ he’@)I < 1
(5.259)
and repulsive if
11
+he’(a)I > 1.
(5.260)
A simple criterion for r > 1 and h sufficiently small may be obtained by replacing (5.257) by i(1)= -E(t)‘,
r!
E(0) = Eo.
(5.261)
Thus (5.262) consequently, a is attractive if d’)(a) < 0 and repulsive if &)(a) > 0 when EO > 0. If 0 then the same conclusion is reached for r odd; however, the conditions are reversed for r even. These conditions for r > 1 are summarized in Table 2. Consider the following examples:
Table 2 Classification of Critical Points for r > 1
Attractive
The First-Order Difference Equation
h < 2, a! is attractive.
hence, for 0 2. e(z) = c(z a!
145
- c),
c
0.
= 1, 3, e‘(1) = c(1 - C), e‘( 0 =$ a! = 1 is repulsive for sufficiently small h. 3. e(z) = Z(I - 22).
1 (supercritical case). It is clear that there is just one number f E (0, 1) satisfying @(C) = 5. This is taken for c in g(z). Using (5.175) or, equivalently, (5.216), one obtainsthe following approximation, 2(t;z ) to ~ ( tz):; (5.266)
Chapter 5
146
The mean number in the population, m(t), is given by (5.267) Let m = m(1); then, since #(z) is the exact one-step generating function, one may take m = q5'(1), hence the final approximation is
m = #'(l), z - 5 - ( z - 1)gm' Z ( t ;z ) = z - ( - ( z - l)m' ' m(t) = m!, In m g(z) = -(z - l)(z - 5). I-< -
(5.268)
Case 2: J(1) < 1 (subcritical case). It will be assumed that #(z) may be analytically continued so that a value
5 > 1 exists satisfying +(c) = 5. The smallest such root is chosen and used in g(z). Again, setting m = r$(l), one finds exactly the same solution, which is given in (5.268). Case 3: #'(l) = 1 (critical case). One may take g(z) = c(z -
(5.269)
yielding
2(t;2 ) = 1 +
z- 1 1 - (z - 1)ct'
(5.270)
(5.272) Let the probability distribution generated by +(z) be p 0 , p 1 , . . ., then (5.272) ensures that 2(t;z ) generates the same value of p o . It is clear, from the construction of the approximations, that 2(t;z ) is exact when #(z) is a bilinear form.
Example: Let B(x) be the distribution function of a nonnegative random variable with Laplace-Stieltjes transform &S), i.e.,
Equation Difference Hrst-Order The
147 (5.273)
then, since &S) is completely monotone, it follows tha: @(z)= &h - hz) (A =- 0) is absolutely monotone; further, since #(l) = B(0) = 1, #(z) is a probability generating function that maybeused to define a branching process. If h is the arrival rate of a Poisson stream and B(x) the service time distribution of a queue, then m = #'(l) is the offered load p; the known stability condition, p < 1, implies the subcritical case, which further implies m(t) -+ 0, t + oo.The function Z(n; z ) (n = 1,2, , , .) is the probability generating function of the number of arrivals after n consecutive services. For the choice B(x) = 1 - e-@', the approximation is exact, in this case, = l / p .
IO. A PERTURBATION SOLUTION OF
A Z =e(z> h
It will be assumed that Z(t; zlh) is analytic in h about h = 0; further, O(z) is taken to be independent of h. Terms of the perturbation expansion in powers of h up to h' willbe obtained. This can also provide significant information concerning the dependence of Z on t and z . Accordingly, let Zo(t;z ) = Z(t; Z I O ) ,
(5.274)
so that Z = Z o + h Z 1 + yh2Z z + .
(5.275)
with the initial conditions
z, = z , (5.276)
2, = 0, Z2 = 0, etc.
the corresponding expansion for O ( Z ) is
e(z)=e(z,) + hz,e'(z,)
+ hy' [Z,S'(Z,)+ z:e"(z,)l+
(5.277)
''*
The difference equation and initial condition may be written in the form
Chapter 5
148
(5.278)
(5.279)
I’
+ h[ ZIe’(Z0)du - f e(Z0) + f + h2[iJi{ Z ~ ~ ’ ( Z+OZ$”(Zo)} ) du - iz,e’(zo) + he(zo)-he(z)]
(5.280)
+ ....
Equating corresponding powers of h in the expansions of (5.275) and (5.280) and subsequent differentiation yields the following differential equations:
io= e(zo), i,= zlB‘(zo) - jzoe’(zo), iz= 4z2e‘(z0) + i z:e”(zo)
(5.281)
- +ile’(zo) - $zlioe”(z0) +&zoe’(zo). Defining M ( z ) by (5.282) one may verify the following solutions of (5.281):
(5.283)
The First-Order Difference Equation
149
Thus the perturbation solution, (5.275), is now
(5.284)
+& IB'(Z0) - d(Z)+ M(Z0) - M ( z ) } ]+
* *
.
Applying (5.284) as far as the first power of h to the examples of(5.243) and (5.248), one obtains respectively
+
Z(t) =
h
+...
ln(l+$)
4 J m
7
(5.285)
2
Z
Z ( t ) = -1 + t z h(&)
ln(1 + tz) +
,
An additional example is provided by A z = e-', h
Z(O)= z
(5.286)
for which one easily obtains Z(t)= z
+ ln(1 + te-")
h +-e 2
+
In( 1 te-") 1 + te-"
11. HALDANE'S METHOD FOR
+ . * m .
(5.287)
AZ = O(Z) h
The final method of solution to be discussed isdue to J. B. S. Haldane [8]. It consists of determining the function F(z) of (5.240) or, equivalently, by (5.242), g(z)-'. One has F(z) = [g(w)-' F(z
dw,
+ hO(z)) - F(z) = h.
(5.288)
Let (5.289)
then, from (5.288), (5.290)
Chapter 5
150
For the integrals in (5.290), one may write (5.291) hence,
(5.292)
Equating corresponding powers of h provides the following formulae: 1 I
c S=
1
(5.293)
qZ)I-s+l
s!(l - S
+
,f,(l-qz) = 0, 1).
l 2 2.
(5.294)
(5.295) Examples are: 1 1. e(z) =-, Z
The First-Order Difference Equation g(z)" = z
151
h h2 ----+...,
22 423 F(z)=Az2--lnz+-+.. h h' 2 2 8z2 ..
1
h
h2E
g w ' = -2+ - + " z + . * *
1 h' E F(z)=-+hlnz+-z+.-. EZ 2 3. e(z) = e", ha - h2a2ea' + . . . g(z)" = e-@' + 2 6 h2ae, + , , . 6 The determination of Z may be accomplished either by solution of the system (5.174), namely
F(z) = --e1
-a2 + Tha Z"
Q ,
Z(0) = z
2 = g(z),
or by solution of the finite equation [see (5.175)] F ( Z ) = F(z) t.
+
(5.296) (5.297)
12. SOLUTION OF G($(z))- /(z)G(z)= m ( ~ ) . Having discussed the nonlinear difference equations (5.171) and (5.172), it is possible now to return to the original equation (5.164). Define Z ( t ) as in (5.171); then the functional equation may be written as G(Z(t
+ 1)) - I(Z(t))G(Z(t))= m(Z(t)).
(5.298)
Let U(t) = G(Z(t)), 4 t ) = I(Z(t>), b(t) = m(Z(t)).
Then (5.298) takes the form U(t + 1) - a(t)U(t)= b(t), which was discussed earlier. Of course, one now obtains G(z) from
(5.299)
(5.300)
152
Chapter 5 G(z) = U(0).
(5.301)
It is now possible to solve the functional equation (5.170) for the'M/M/I queue with a feedback customer. Defining Z ( t ) by Z(t
+ 1) = 1 + p( 11- Z(t))'
Z(0) = z
(5.302)
and using (5.195), one finds Z(t) =
1 - pz - (1 - z)pf 1 - pz - (1 - ,)p'+'
(5.303)
'
Setting (5.304)
the functional equation becomes U(t
+ 1) - a(t)U(t)= 0,
U ( m )= 1.
(5.305)
Theboundary condition follows from Z(a0) = 1 , G ( l ) = 1. Thus, the required solution for U ( t ) is I
Slna(w)Aw.
U ( t )= e m hence,
(5.306)
I
O
S a(w)Aw.
G(z) = e m
I
n+
(5.307)
W
=
[l
up(1 - Z(j))]-l*
j=O
Y
A perturbation solution for G(z) when Y is near one is readily obtained; let = 1 E , then
+
a(t) = 1
+ p(1 - Z ( t ) ) + EP(1 - Z(t)),
(5.308)
and 0
G(z) = exp[S In( 1 W
+ p(1 - Z(w))Aw] exp
(5.309)
Difference The First-Order
153
Equation
A simple general formulation for the solution of the equation G(#(z)) - G(z) = 4 z )
(5.310)
is obtained as follows. The substitutions (5.299j lead to A V(t)= b(t)
1)
(5.31
and, hence, to t
G(Z(t))= K ( z )
+ $m(Z(w))Aw.
(5.312)
Differentiation with respect to t at t = 0 yields
+ H0
G’(z)g(z)= ~ ( z ) m’(Z(w)>Z(w)Aw
13)(5.3
which permits determination of G(z). Consider the example (5.314) One has Z
Z(t)= 1 tz’
g(2) = -22;
(5.315)
1 Awn 0 (1 wz)2
(5.316)
+
hence 1 0 G’(z) = -;+S
+
The change of variable W = y / z gives 1 l0 1 G’(z) = --+-SA Y’ z ZO(l+y)2z and the further change
W
(5.317)
+
= 1 y gives
1 11 1 G’(z) = - - + - S - - W .
(5.318)
From Chap. 3 (3.74), one has X1 $(XlZ) = W,
(5.319)
2
Z 1 W 2 I
S; 4
hence (5.320)
..
, .
.. ..
.
....
.
..._
.“
........ ..
..
.
I .
.
... .
154
Chapter 5
Since, from (3.78), (5.321)
one gets (5.322)
and, finally, G(z) = $(L). Z
(5.323)
13. SIMULTANEOUS FIRST-ORDER EQUATIONS Here, the theory of the first-order nonlinear difference equation is extended to a system of simultaneous equations. In what follows, it will be useful to x,) with scalar components use the notation 2 for the vector (21, 2 2 , 1 5 i 5 n. The general form of the system to be discussed is a ,
A zj(t; 3 = e,(i), ~ ~ (30=; z j , h
1 5 i 5 n,
(5.324)
Z’=(z1,zz,,**,z,),
in which ei(3 are independent of t . Define the infinitesimal generators g,(3 by (5.325)
then, because of time homogeneity, the followingsystemof equations is equivalent to (5.324): dZi -= gi&, z ~ (=ozi,) 1 5 i 5 n. dt The consideration
+
Z*(t S ) ; 2) = zi(t;i ( S ; Iv
a),
zj(t;i+ ;S>,
differential (5.326)
(5.327)
S “P 0
shows that the partial differential equation (5.328)
Equation Difference First-Order The
155
is also equivalent to (5.324); the solutions of (5.328) are the n functions Z,(t;2). In order to construct the solution of the system (5.324), a U-operator approach will be followed. The total difference quotient with respectto t, A, implies h Af(&)) h
= i l f ( i ( t + h)) -f(%))l.
(5.329)
Setting wi
= h8i(2),
Zi(h) = z j + w j ,
(5.330)
one has
hence W(z3 = +f(&))Ir=o.
(5.333)
Clearly, one also has
in which Uois taken to mean the identity operator. It is possible to put the definition of U into another form that is suggestive of partial differentiation [8].Define A by
Af(3 =f(; + 2)-f(2), in which the wi are treated as constants; then A =E? ...E," - 1;
(5.335) (5.336)
the Ej are translation operators each referring only to the respective zi. Define Ai to be the partial difference quotient operating only with respect to zi, th&
(5.337)
156
Chapter 5
(5.338)
(5.339) 0 3
+ en(z ) Anf(zl + ~1 2
I
zz
+W ,
I
zn-1
+ o n - l , zn).
W,
The partial difference operations are carried out with wi constant and then their values are assigned as given in (5.330). Newton's expansion is now used to express the solution of (5.324); thus,
(5.340) j=O
In symbolic form this becomes
+
f ( 3 = (Z hU)"hf(?).
(5.341)
Differentiation with respect to t at t = 0 and use of (5.328)yield
(5.342) or, in expanded form,
In particular, choosingf to be a function only of zi, one has
(5.344) j=O
The First-Order Difference Equation
157
and
(5.345) These expansions may be used in the same manner as (5.225) and (5.251). If limiting forms exist for h + 0+, the previous formulae become the well-known Lie-Grobner formulae [4]for the solution of a system of differential equations; thus,
dZi = ei(,?), dt
Zi(0)= zi,
1 5 i 5 n,
(5.346)
The last expression for U is the limiting form obtained from (5.339). A function G(2) satisfying
(5.347)
UG=O
is called an invariant function of the difference equation system. It is determined by
(5.348) and, from (5.340), satisfies G ( i ( t ) )= G($
(5.349)
The related function F ( 3 given by
(5.350)
UF(3= 1 may be determined from
(5.351) and satisfies F ( i ( t ) ) = F(,?)
+ t.
(5.352)
These functions provide useful insight inta the nature of the solutions for example, conservation type results.
zj(t);
158
Chapter 5
The nth order equation
may be rewritten as an autonomous system of first order by setting Zi(t) = Z ( t
+ (i - l)h),
Z,+l(f> = t;
1 Ii In,
(5.354)
thus, one has
(5.356)
(5.357)
(5.358) Z,(t) = z2
+ t.
Example 2 Z(t + 2) - 5Z(t + 1) + 6Z(t)= 0. This equation is more directly solved by the methods of Chap. 6; however, it willbediscussed here as an illustration of the present method and as an example of .the divergence of (5.344). Let Zl(t) = Z(t), Z2(t)= Z(t + l), then
Equation Difference First-Order The
159 A21 = 2 2 - 21, A22 = 422 - 621, el = Z, - z l ,
(5.359)
e, = 422 - 621.
Clearly, Ujzl has the form UJzl = ajzl
+ bjz2;
(5.360)
hence, from d"z1 = U(Ujzl), the following matric equation is obtained: (5.361)
with the initial values a. = 1, bo = 0. Since the eigenvalues ofthe matrix are 1,2, aj has the form aj=A+B2',
(5.362)
hence aj = 3 - 2 * 2 ' , bjZ2'-l,
(5.363)
and UJZ, = (3 - 2 2')q
+ (2' - l)zz,
j
L 0.
(5.364)
Substitution of UJz,into (5.344) yields a divergent series; however, using the expansion
2 (f)"
= (1 +a)'
(5.365)
j=O
as though it were valid for a 2 1, one gets the correct solution, namely zl(t) = (321 - 22)2' i(Z2 - 221)3'.
(5.366)
The foregoing procedure can, in fact, be justified by use of summability methods [37]. The Euler (E,q) method is particularly suitable because it sums a power series beyondits circle of convergenceand may also be useful for numerical computation. Let n
S,, =
C aj
(5.367)
j=O
be a given series and let q > 0 be chosen; then ay), siq) are defined by
Chapter 5
160
(5.368)
n
m
(5.369)
The case of convergence corresponds to q = 0. An example is given by the power series aj = d . One has (5.370)
hence, the series is summable in the circle (5.371)
Iq+xl < q + 1 ,
+
that is, with center -q and radius q 1. The following holds for summability (E,q) [37]: (5.372)
This property is called consistency;the special caseq = 0 is called regularity. Thus, every convergent series is summable (E, q) to the same s u m . The power series example of (5.730) may be generalized. Let
f ( x )=
0 3 . C bjx’,
(5.373)
j=o
uj = b i d , and let ( be the singularity of f ( x ) nearest the origin; then the power series is Euler summable within the circle
+
145. x1 < (4
+ 1)l5.l*
(5.374)
This may be applied to the Newton’expansion
2 (f )d
= (1 +x)‘
(5.375)
j=O
for which ( = -1; hence, the expansion is summable in the circle ( x - q l < q + 1.
(5.376)
First-Order The
161
Difference Equation
Applying Euler summability to (5.344) on substituting (5.364) and using (E, l), the solution given in (5.366) is obtained. Also, in this case, the infinitesimal generators are easily found, namely 3 8 g1(z1, z2) = z1 In - z2 In - , 9 2 2 27 g2(z1,z2)=6z11n-+z2ln3 4'
+
(5.377)
Example 3: The Erlang loss function (Chap. 2: M/M/c Blocking Model) satisfies the equation (2.26) t+l ~ ( t1, U)-' = - ~ ( t ,
+
U
a)-'
+ 1.
(5.378)
From the point of view of numerical accuracy obtainable from a given number of terms of (5.344), it is sometimes advantageous to transform the dependent variable. In this case,becauseof the exponential behavior of B(t, a), it is better to treat lnB(t, U ) . Accordingly, setting Z l ( t )= lnB(t, a),
Z2(t) = t ,
(5.379)
one has
(5.380)
A22 = 1,
Thus,
(5.381) 2 1
= 21
+ tUz1+- t(t -2 l) U2Zl + . .. *
In fact, this solution finds much use in teletraffic network studies and in realtime traffic systems. A useful,simple approximation to the infinitesimal generators, g@), 1 f i 5 n, may be obtained from
Chapter 5
162 t
zi(t)= zi + S0 e,(Z(u))Ah U -
0
e,(i(U))A U. h
(5.382)
Differentiation with respect to t at t = 0 yields (5.383)
hence, approximately, (5.384)
It is useful to introduce the matrix M defined by (5.385)
in which I is the identity matrix. Defining the column vectors (5.386)
the system of equations (5.384) takes the form
MG = 8. (5.387) Thus, solution of (5.387) provides approximations to the generators gi. Equation (5.387), in fact, subsumes the approximate derivative formula of (5.154), which may now be seen to be an approximate construction of the infinitesimal generator for the complete equation (5.27) when written in the form (5.151).
PROBLEMS 1. Solve
+ 1) - eZXu(x)= 0, ( x + 2)u(x + 1) - 2(x + l)exu(x) = 0, u(x + 1) - u(x) = x sin x , u(x + 1) - xu(x) = xpx, 2 u(x + 1) - eZXu(x)= xe u(x
,
The First-Order Difference Equation
163
2. Solve
a
+ l)u(w,x ) - wu(w,x + l), x ) = (W + x - a)u(w,x ) + wu(w,x + l), x ) = -(x + l)u(w, x ) + ( x + l)u(w, x + 1). x ) = (W
W"(W,
aw
a
W-u(w,
aw
a
W"(W,
aw 3. For the Erlang loss function, B(x, a), let B, = aB/ax, B,, = a%/ax2, r2
= B,,/B,,
a!
r1 = B,/B,
+ 1 - r i1.
=X
Obtain the approximation rz = 2r1
+ In(a!/a) 1 + 1/(2a!r1) Q-'
*
4. Show that for any functions, m(z), n(z),one has
+ hO(z))Vn(z)+ n(z)Um(z). 5. Show that for Z(t + 1) = o Z ( ~ ) Z(0) ~ , = z, one has U[m(z)n(z)]= m(z
g(z) = z In(az) In 2, In In(az) F(z) = In2 ' 6 . Solve approximately (in all cases Z(0) = z):
A z = &'-l)
- Z.
h
7. Show that the mean, m, of the distribution generated by G(z),(5.170), is m = - rP 1- p '
8. For G(z) defined by (5.310), show
Chapter 5
164
9. Using the Haldane method, the solution of
UG=m may be obtained in the form
Obtain the following formulae for thef,(z). m
f2
1 1 2,
f,'""'(z) = 0,
= -8"-, d m dz 8
d2 m
h
10. Let &(z) be the nth iterate of $(z), and let
F(2, W ) = z
+ wqqz) + w2&(z) +
* ''
;
then show F(z, W ) = z
+ wF((b(z),
W).
+
11. Let Z(t 1) = $(Z(t)), Z(0) = z; show that the function F(z, W ) of Prob. 10 satisfies
F(z, W ) In W
a + &(F(z,
w))g(z)= -z
- S00 w"[Z(u)
In W + Z(u)]Au.
12. The equation A 2 = O(Z), Z(0) = z may be rewritten in the form h
hence, obtain the approximation
13. Solve
G(z) = G(-)C?'(~-~), 2 -12
The First-Order Difference Equation
165
15. Solve G
($"
- G(z) = z.
16. Obtain the solution
of G(z2) - G(z) = Z,
z > 0, z # 1.
17. For the equation AZ=eaz-l-Z, h
Z(O)=z,
1 l - % c a = ro(1 - W))
The First-Order Difference Equation
169
where $(z) = e(&
If h@
- az)).
1 and Z(t
+ 1, z ) = $(Z(t, z ) ) with Z(0,z ) = z , show that
m
j=O
28. Define & ( t ) andf(z) by i o ( t )= e(zo(t)),
zo(o)= Z ,
f’(z)e(z)= 1;
Show that the solution of A z = e(z),
z(o)= z
h
is given by
Thus, an approximation to Z ( t ) is, for example,
z(t)= Zo(tUf(z))29. Obtain the following solution of UG = m : G’(z)g(z)= m(z) -
00
CL,hUU’”’ [m’(z)g(z)] v= l
in which the L, are Laplace numbers. 30. Consider the following method ofsuccessive approximations for the solution of
z(t)= e(z(t)), z(o)= z . Define the sequence Zj(t) by &(t)
+t I
Zj+l(t) = Z
e(zj(W))
2W -
z and
0
e ( z j ( w ) )A W , h
. I ’ L 0;
170
Chapter 5 then the function Zj(t) is taken as an approximation to Z ( t ) . Show that for the equations
I
AZ=Z’
f(0) = z ,
one has, respectively, Zz(t) = z
+ t -l +Z ’ [ @ (
(1
+ z)2 hz + tz
) -@(v)]’
3 1 . Determine the character of the critical points of the following equations: A z = z(z- 1)’ h
A z = z2(z A Z = sinZ - Z . h
32. Consider the branching process for which B(x) = 1 - e-@’ (section on branching process approximation). Show
p - l “f -1 - -
P&)
P
p z p n
33. Let a! be an attractive critical point of A z = e(z), h
Z(O)= z .
+ Z ( t ) a! + c(z)a”h, t C ( Z + hB(z + a)) = ac(z)
Let a = 1 “(a!); show
-
00, (Schroeder equation)
?,
34. Consider
z-1
AZ = Z 2-2’ Show
Z(0) = z.
The First-Order Difference Equation Z(t)
-Z
2-',
1-z
Z(t> -- 2 -l-Z(t)
1-z
t+m,
171
O 1.
"n This approach provides
I
Figure 2
,. .,.,.
,,
,
.,...,_,.,
.
.
.
I
.
,..... . . . . . .
.
.
.... .. ,... ..
....... . ,
. .,
Chapter 6
188
(6.108)
4. EQUATIONS REDUCIBLE TO CONSTANT COEFFICIENTS The following class of equations is reducible to the constant coefficient case: u(x
+ n) + an-l@(x)u(x+ n - 1) + Un-2q5(X)f$(X - l)u(x + n - 2) +
+ aoqw#@ - 1)
' 4qx
- n + l)u(x) = &?(x).
'''
(6.109) Let O(x) designate a solution of AO(x) = In #(x
+ l),
(6.110)
for example, X
e(x) = S lnq5(z
+ l)Az,
11)
(6.1
and let u(x) = ee(x-n)v(x),
12)
(6.1
then . v(x
+ n) + an-lw(x+ IZ - 1) + - + aow(x) = e-e(x)g(x).
(6.113)
An example is given by U(X
+ 2) - 5 x 4 +~ 1) + ~ X ( -X l)u(x) =(6.1 1.14)
Here, one has 4(x) = X , e(x) = In r(x U ( X ) = r(x - I)v(x), V(X
+ 11, 1
+ 2) - ~ W ( +X 1) + ~ v ( x =) r(x + 1)
(6.115)
Using Broggi's method (6.74), the particular solution obtained is (6.116)
The Linear Equation with Constant Coefficients
189
5. PARTIAL DIFFERENCE EQUATIONS The difference equation in more than one independent variable, e.g., Lu(x, y ) = 0, will be discussed here by means of Boole's operational method, Lagrange's method, and the method of separation of variables; more information is available in [Refs. 8, 9, and 341. The operators E , A will be subscripted to show the variable to which the operator refers. Boole's method is illustrated in the following examples.
Example 1. Lu = u(x + 1,y ) - au(x, y One may write u(x
+ 1) = 0.
+ 1,y ) - aEyu(x,y ) = 0;
(6.117)
hence, treating Ey as a constant, u(x, y ) = aXE;c(y) = d C ( y
+ x);
(6.118)
here an arbitrary function takes the place of the usual arbitrary constant. One may also introduce the arbitrary periodic P(x, y ) of period one in each variable and write
+
u(x, y ) = P(x, v)aXc(y x);
19) (6.1
however, this will be omitted in the succeeding examples.
Example 2. Lu = u(x One has
+ 1, y + 1) - u(x, y + 1) - u(x, y ) = 0.
+ 1,y ) - Eyu(x, y>- 4 x 9 v) = 0,
+ 1,v) - (1 + E,-l)u(x,y ) = 0 , u(x, y ) = (1 + E,-l)xc(y), u(x
(6.120)
j=o
This is not the only form of solution of this equation; one may also write E,u(x, y #(X,
+ 1) - u(x, y + 1) - u(x, y ) = 0,
v) = A i y @ ) .
(6.121) (6.122)
Pascal's triangle is contained in this equation. Consider the boundary conditions u(0,O) = 1, u(0, y ) = 0 0,# 0); then c(0) = 1, c(y) = 0 (y # 0) and, hence, u(x, y ) =
c).
Chapter 6
190
Example 3 Bernoulli trials:
+ 1,y + 1) = pu(x, y ) + qu(x, y + 11, + q = 1, p 2 0, 4 2 0, u(0,O) = 1, u(0,y) = 0 0,# 0).
u(x
p
One may interpret x a s the total number of trials and y as the number of successes with p the probability of success in any one trial. One has
(6.123) j-0
The boundary conditions imply c(0) = 1, c0,) = 0 0,# 0), hence y ) = &-ypY,
(6.124)
that is, the probability of y successes in x trials. It is seen that the operational method reduces a partial difference equation in two independent variables to anordinary equation containing operator coefficients. Clearly, variable coefficients may occur in the variable to which the operator does not refer.
(6.125) The method of Lagrange applies to equations with constant coefficients; it consists in assuming a solution of the form 4x9 v) = dsy.
(6.126)
A relation is established between Q , # so l that one constant may be eliminated. Example 5 Lu = u(x Let u(x, y ) = aXs’;
then
+ 1,y + 1) - u(x, y + 1) - u(x, y ) = 0. (6.127)
The Linear Equation with Constant
Coeficients
191
ap-/5-1=0, (6.128)
a = B"l(1 + B ) ,
and, hence,
+ B)xc(B)
u(x, Y ) =
(6.129)
in which c@) is an arbitrary function of /3 . Hence also (6.130)
+
Example 6 Lu = u(x 2, y ) - 02u(x, y Here, one has two values for Q
= op-1i2,
- 1) = 0. (6.131)
-ap"i2,
hence
in which C(z), D(z) are arbitrary functions.
+
Example 7 u(x 2, y ) - 2u(x The resulting equation
+ 1, y + 1) + u(x,y + 2) = 0.
(a- p)2 = 0
shows that B"+y,
a!
=
(6.133) is a double root, hence (6.134)
xp+y
are independent solutions; it follows that u(x, y ) = c(x
+y ) + xd(x +U)
(6.135)
in which c(z), d(z) are arbitrary functions. The technique of separation of variables consists of substituting 4x9 Y ) = fWBO.1)
(6.136)
and reexpressing the equation so that a function only of x appears on one side of the equation and a function only of y on the other side. Each side may then be equated to an arbitrary constant, thus providing two decoupled ordinary difference equations for the determination of @(x),Bb).
Chapter 6
192
Example 8 Same as Example 4. One has a(x
+ 1)Bo,> = axa(x)Bo, + 11, + 1)
Bo, + 1) - Y. - BO)
(6.137)
” ”
Thus the two equations are a(x
+ 1) = yaxa(x),
(6.138)
Bo, + 1) = YBO). Since the solutions are .(X)
= faxr(x),
(6.139)
Bo,) = r‘ one has
v) =
L
U”+YC(r) dY,
(6.140)
= axr(x)c(x +V)
which agrees with (6.125). Example 9 Stirling equation. Equation (1- 31) satisfied by the Stirling numbers of the second kind can be written in the form u(x+l,y+1)=(x+l)u(x+1,y)+u(x,y).
(6,141)
Use of (6.136) and separation of variables yield
(6.142)
Thus one has
(6.143)
The Constant Equation Linear with
Coeflicients
193
X
To obtain the Stirling numbers, S , one restricts x, y to integral values; Y hence the following sum may be considered: (6.144)
The boundary condition to be satisfied is
si =o,
x > 0,
= 1,
(6.145)
x = 0.
The Newton null series
c X
= 0,
x > 0,
=l,
x=o
(6.146)
i=O
provides the required key. Setting (-1)' c(i) = i!
(6.147)
now yields the explicit formula (6.148)
+
Laplace observed that if in Lu,x y , or x - y , is constant in the arguments of U in each term, then the equation can be reduced to ordinary form. Let, for example, x y = a; then the substitution
+
u(x, y ) = y(x, a - x) = v(x)
(6.149)
results in an ordinary difference equation for v(x).
+
+
+
Example 10 Lu = u(x 1 , y 1) - (x l)u(x, y ) = 0. In this equation y - x is constant in each term, hence setting y-x=a, u(x, y) = u(x, a
(6.150)
+ x ) = v(x)
one has v(x + 1) - (x
+ l)@)
(6.151)
= 0.
Thus
,,..
. ..
,..,, " .
...I . .
.
194
Chapter 6
(6.152 Example 11 (Boole) [43]: A and B engage in a game, each step of which consists of one of them winning a counter from the other. At the commencement, A has x counters and B has y counters. In each successive step the probability of A's winning a counter from B is p , and therefore of B's winning a counter from A is q(p + q = 1). The game is to terminate when either of the two has n counters. What is the probability of A's winning it? Let u(x, y ) denote the probability A wins starting from state (x,y). If A gains a counter (with probability p ) , then the state becomes (x + 1, y - 1) and u(x 1, y - 1) is the probability A wins thereafter. Also if A loses a counter (probability q ) then u(x - 1, y 1) is the probability A wins; hence the required difference equation is
+
+
u(x,y) = p u ( x + 1,y- l)+qu(x-
l,y+(6.153) 1). It is observed that x + y is constant in each term. Let the total number of counters be a, then x+y=a, u(x, y ) = u(x, a - x ) = w(x),
pv(x
(6.154)
+ 1) - w(x) + qw(x - 1) = 0.
Let y = q / p ; then the solution is 'u(4 =c W , y # 1, = c dx, y = 1, hence
+ +
U(X,Y)=c(X+Y)+d(x+y)yX, =c(x+y)+d(x+y)x,
(6.155)
v#
1, y = 1.
(6.156)
The boundary conditions are u(n, a - n) = 1, u(a - n, n) = 0; hence c(a>+d(a)y" = 1, c(a)
+ d(a)yQ"" = 0,
c(a)+d(a)n = 1, c(a) d(a)(a - n) = 0,
+
y # 1, y # 1, y = 1,
(6.157)
y = 1.
The required probability is now
(6.158)
Equation Linear The
with Constant Coefficients
195
A player is saidto be ruined if he loses all hiscounters; thus, setting n = a yields the probability that player B is ruined. One has P(B is ruined) = y“-l’ X
=n’
yZ1, (6.159)
y = l.
Example 12 (Finite Source Model): The finite source model to be discussed [26] consists of n sources and one server. The service rate is W,With rate y , a source is expectedto generate a request for service; after a request is placed, it cannot generate further requests until the required service is completed. Requests are held in a first in, first out queue awaiting start of service. A source that can generate a request is called “idle” and is said to be thinking; the mean think time is y-l. A source that has placed a request is termed “busy’; the mean waiting time, W , is the time from initiation of the request until the start ofservice. The mean servicetimeis p-’. The total mean request rate over all time is designated A; thus, A/n is the request rate per source and n/A is the mean time between requests. The relationship between these mean times is shown in Fig. 3. The following conservation relation holds W
+ p-’ + y-’
= na-’.
(6.160)
To analyze the system in equilibrium, let x be the number ofbusy sources, y the number of idle sources, and let u(x, y ) designate the probability the systemisin state u(x, y). Considering the neighboring states (x + 1, y - l), (x - l , y l), the following rate equation may be written:
+
( y y + ~ ) u ( x , y ) = y O / + l ) u ( x - l , y + l ) + ~ u ( x +1,y-l),
0 < x < n,
(6.161)
request
end service
Start
service
Figure 3
next request
Chapter 6
196
in which the left side isthe rate of leaving state (x,y ) and the right side isthe rate of entering (x,y ) from the neighboring states. The boundary conditions at x = 0, n are ynu(0, n) = pu(1, n - l), (6.162) pu(n, 0) = yu(n - 1 , 1). Since x + y = n, one may introduce
w(x) = #(X,n - x).
(6.163)
Rewriting (6.161), (6.162) and setting 2 = y / p , one has
+
[2(n - x) + l]w(x) = 2(n - x l)w(x - 1) + w(x + l), (6.164) 2nw(O) = w(l), w(n) = 2w(n - 1). Observing that w(x + 1) - 2(n
- x)w(x) = w(x) - 2(n - x + l)w(x - l),
(6.165)
it follows that
w(x
+ 1) - 2(n - x)w(x) = c
(6.166)
in which the constant c may be evaluated from c = w(1)
- linv(0).
(6.167)
The boundary condition shows, however, that c = 0; hence one has the firstorder equation
w(x + 1) - 2(n - x)v(x) = 0.
(6.168)
Since, for this model, x is an integer, the appropriate solution of (6.168) and (6.164) is
w(x) = An(X)?,
0 < x < 12.
(6.169)
The value of A follows from
c n
w(x) = 1
(6.170)
x=o
and hence
A=
[2
n(’)?]
-1
= B(n, 2-l)
(Erlang loss function).
(6.171)
The identification with the Erlang loss function stems from (5.72). The required probability distribution of busy sources is now
Constant Coefficients Equation Linear with The
197
v(x) = n(")h"B(n,h-').
(6.172)
Since the rate of requests entering the queue must equal the rate leaving the server, one has h = (1 - B(n, h-'))p,
(6.173)
hence the total offered load a = h / p is a = 1-~
( nh-'). ,
(6.174)
Let L be the mean number of busy sources and I that of idle sources; then L + I = n. One has h = VI,
(6.175)
hence (6.176) The mean waiting time, write W
= LA" - p -
1
W,
is simply obtained from (6.160). One may also
.
(6.177)
A particular solution of the inhomogeneous form Lu(x,y ) = g(x,y) may be obtained by the same methods employed earlier for the ordinary equation Lu(x) = g(x).
Example 13. Lu = u(x + 1, y ) - au(x, y solution is given by
+ 1) = bX(x+U).
A particular
(6.178) in which the shift theorem was used. Thus U
1 b-a+bA,-aA,
= bX
(x +
v) (6.179)
For the case b = a, one gets
Chapter 6
198
(6.180)
Example 14 Lu = u(x
+ 1 , y + 1 ) - u(x,y + 1 ) - u(x,y ) = 2x3y. One has
2x3y ExEy - Ey - 1 1 = 2x 2ExEy - Ey - 1 1 = 2'3Y l 6ExEy- 3Ey - 1 = 2'"3Y.
U =
3y
(6.181)
The solution obtained in (6.130) may now be added to the particular solution to form the general solution.
PROBLEMS 1. Solve U(X
+ 3 ) - ~ U ( X+ 2) + 2
6 ~ +( 1) ~ - 2 4 ~ (=~ 0.)
2. Solve U(X
+ 3 ) - ~ U ( X+ 2) + 26u(x + 1) - 24u(x) = 0 .
3 . Solve U(X
+ 3 ) - ~ U ( +X 2) + 27u(x + 1) - 27u(x) = 0.
4 . Solve
u(x + 2 ) - 3u(x + 1) + 2u(x) = x23x. 5. Solve using the Norlund sum u(x 2 ) - (a B ) U ( X 1 ) a/?u(x) = g(x). 6 . Solve using Broggi's and Laplace's methods 1 U(X 2) + ~ U ( X+ 1 ) + 6 4 ~=) x2 * 7 . (Boole) A person's professional income is initially $a which increases in arithmetic progression every year with common difference %b.He saves
+
+
+
+ +
Constant Coefficients Equation Linear with The
199
l/m of his income from all sources, laying it out atthe end of each year at r percent per annum. What will be his income when he has been x years in practice? 8. There are only two states of weather, fair and poor. The probability on day zero of fair weather is p and the probability that the weather on day n 1 continues the same as on day n is q? What is the probability, p , that the weather is fair on day n. 9. Using (6.98), evaluate the following:
+
X
F(xlw) = S e-orrsinz A 2 , 0
W
(erfcx is the complementary error function) 10. Solve for the eigenvalues h, and eigenfunctions u,(x)
A%(x - 1)
+ A.u(x) = 0 , ~ ( 0 =) 0, u(N + 1) = 0 ,
X
= 1,
. . . ,N .
7 Linear Difference Equations with Polynomial Coefficients
1. INTRODUCTION This chapter presents methods for the solution of linear difference equations with polynomial coefficients and applications to two queueing models. The next section discusses the technique of depressing the order of a difference equation when at least one solution of the homogeneous equation is known. For the caseof the second-order homogeneous equation, the useof Casorati’s determinant and Heymann’s theorem isshown to provide the second solution. Of the many methods that can be used to solve a difference equation with variable coefficients, expansion into factorial series of firstand second kinds appears to be of broad applicability. The n and p operators introduced by George Boole [43] and furtherdeveloped by Milne-Thomson [8] are studied. These are particularly useful in obtaining factorial expansions. Application is made to difference equations of specialized forms expressible solely in terms of either p or n. Application is made in Section 4 of the x,p operators to the general homogeneous equation. A procedure is discussedthat permits reduction to a canonical form from which the factorial series expansions of the solutions are obtained. Some exceptional cases arise when the roots of the indicia1 equation are zero, multiple or differ by an integer. The relevant methods of solution are introduced, and the complete equation is solved by 200
Polynomial Equations with Linear
201
Coefficients
means of expansion of the inhomogeneous term in series and also by use of the Lagrange method of variation of parameters. The last come, first served(LCFS) M/M/C queue with reneging isimportant in certain teletraffic models, e.g., delayuntil a dial tone is received [44]. The Laplace transforms conditioned on all servers busy of various performance parameters of interest are obtained. The representation of these parameters by means of the transform is then introduced, after which the explicit solution of the model isobtained. We then obtain mean values and a simplification of the waiting time transform that permits accurate inversion for the values of data encountered in practice. The M/M/l processor-sharing queue used, for example, as a model in round-robin computer communication systems is introduced [46]. The state equation is established for the Laplace-Stieltjes transform of the response time conditioned on the number present in the system [6]. This provides a backdrop for introducing the method of singular perturbations, which is then used to solve the problem.
2. DEPRESSION OF ORDER Introducing the operator L by Lu = a,(x)u(x n) a,-l(x)u(x
+ +
+ n - 1 ) + + ao(x)u(x)(7.1) ' ' '
in which aj(x) (0 5 i 5 n) are polynomials, the equation to be studied is
(7.2)
Lu(x) = g(x).
If a solution, w(x), of Lv = 0 is known, the order of L may be depressed.Let (7.3)
u(x) = w(x)t(x); then Lu = a,(x)w(x
+ n)t(x + n) + a,-l(x)u(x + n - l ) t ( x + n - 1)
+ + ao(x)u(x)t(x). '''
(7.4)
Use of Newton's formula
+
t ( ~ Y) =
C (i')Ajt(x)
j=O
in (7.4)results in an equation in which t(x) is absent. This occurs because the corresponding coefficient is LW = 0. Setting At(x) = w(x), an equation of lower order is obtained for w(x). This procedure, of course, is applicable even when the a,(x) ( 0 5 Y 5 n) are not polynomials. One may observe, in particular, that knowledge of one solution of the homogeneous form of a
Chapter 7
202
second-order equation permits reduction to first order of the complete equation and, hence, permits solution by the methods already presented. An example is given by
+ + ( 6 +~ ~)u(x)= g(X)
x 2) - ( 8 -~~ ) u ( x 1) LU= ( 2 -~ ~ ) u ( + for which L(3') = 0. Accordingly, set U(X)= 3't(x),
W(X)
= At(x);
then (18~ 9)t(X
+ 2) - ( 2 4 -~ 6)t(x + 1) + ( 6 +~ 3)t(x) = 3-'g(~).
Using
+ 1) = t(x) + W@), t(x + 2) = t(x) + 2w(x) + Aw(x), t(X
one gets
+
~ 1 2 ) ~ (= ~ 3-'g(~), ) (18~ ~)Aw(x) ( 1 2 -
+
( 6 -~ ~ ) w ( x 1) - (2x
+ l)w(x) = 3-x"g(~).
Instead of solving this equation by the methods already given, the solution willbe obtained by Lagrange's method of variation of parameters, to be discussed later. If the complete solution of the homogeneous equation Lu = 0 of second order is to be obtained and one solution w(x) is known, then an alternative to the method of depressing the order is the use of Casorati's determinant and Heymann's theorem (Chap. 2). Thus, for the operator L of (7.6), let w(x) = 3' and let v(x) be the second solution; then Casorati's determinant is (7.12) Heymann's theorem yields 6x+ 3 D(x 1) = -D(x) 2x - 1 whose solution is
(7.13)
D(x):= 3'(2~ - 1).
(7.14)
+
Combining this with (7.12) provides the equation v(x
+ 1) - 3v(x) = 1 - 2x
(7.15)
Polynomial Coefficients Equations with Linear
203
whose solution is v(x) = x . This is often a convenient method of solution.
3. THE OPERATORS n AND p An operational method introduced by Boole [43] and later developed by Milne-Thomson [8] will be usedto effect the solution of differenceequations in factorial series by techniques that resemble the Frobenius method for differential equations. The definitions of Milne-Thompson will be used for the operators x , p. Let Y be arbitrarily chosen for convenience depending on the difference equation and set x' = x - r, then the definition of p is ,
Thus:
(7.17)
The operator p obeys the index law; thus,
- r(x' + 1) r(x' + 1 - m) r(2+1 - m ) r ( x ' + 1 - m - n ) - r(x' + 1) u(x - m - n) r(d + 1 - m - n)
u(x - m - n)
(7.18)
= pM+"u(x). When the operand u(x) = 1, it is convenient to write just p" so that (7.19)
An expression of the form Newton series:
C,"=,aS/s!pk+' is immediately interpretable as a
Chapter 7
204
F.(" ;
r(x' + 1) W ' + 1 - k) $=o
s=o
k)
(7.20)
Similarly, the sum EgoaSs!pk-' yields a series of inverse factorials:
(7.21) The expansion of functions g(x) in Newton series or series of inversefactorials will be useful in the solution of equations. A monomial equation in the operator p , f ( p ) u ( x )= g(x), has the form aou(x)
+ UIX'#(X - 1) + + a,x'(")u(x - n) = g(x) '
*
(7.22)
in whichf(p) is a polynomial in p. Since pm[r(x'+ I)w(x)] = r(x'
+ I ) V ( X - m),
(7.23)
the substitution U(X)
= r(x'
+ l)+)
(7.24)
reduces(7.22) toanequation with constant coefficients(see Chap. 6). Alternatively, resolution off@)" into partial fractions makes the solution depend on the interpretation of the form (a- p)-k. To interpret (a- p)-', consider (a- P M X ) = g(x), au(x) - ( x - r)u(x - 1) = g(x)
(7.25)
whose solution is (7.26) Repetition of this operation or use of (3.51) will interpret (a- p)-k. Solution of (7.22) in terms of factorial series of the form (7.20) or (7.21) may be obtained by expressing g(x) in factorial series in terms of p and either assuming an appropriate expansion for u(x) in terms of p and equating coefficients or by expanding f(p)". Example: The function satisfied by the Erlang loss function B(x, a) in the form #(x) = B(x, U)-', (5.64), is
au(x) - xu(x - 1) = a. Thus (r = 0)
(7.27)
205
with Polynomial Coefficients
Equations Linear a u(x) = a-p and, using (7.26),
(7.28)
(7.29) see (5.69). Expanding a/(a - p) in positive powers of p yields (7.30) This is a known asymptotic solution (a + 00) [29] useful for the computation of u(x) when a is large. Expanding now in powers of p-' yields W
u(x) = - p p - s = -
c W
a"
(7.31) (x 1) ' ' ' (x S)' This is a very useful convergent representation of u(x)-see (5.67)"especiallywhen a is not large compared with x. This result could also be obtained from (7.29) by setting c = 00. The operator rc is defined by S=
1
S=
1
+
+
4
rcu(x) = x' - u(x) = x'(u(x) - u(x - 1)).
(7.32)
The operation may be repeated so that ff is defined for integral n 2 0. The equation rcu(x) = u(x)
(7.33)
has the solution (7.34)
m-'
hence # r c - l r c unless c is specially chosen. This will always be assumed although the value of c will rarely be needed. Henceforth one will have rc 7c-l = rc-lrc. Clearly, ff obeys the index law withnodesignating the identity for positive and negative n. It may be observed that exactly the same difficulty occurred with the operator A , which, however, did not occasion any problems. An important relation for the present purposes is the following shift formula: f(rc>p"u(x) = Prnf(rc + m>u(x> valid for any rational functionf. To show this, consider
(7.35)
Chapter 7
206
= p"(.
+ m)u(x).
Thus (7.37)
and, inductively, for integral n 1: 0 7Pprnu(x)= p y n
+ m)"u(x).
(7.38)
Because of the obvious linearity of the operators, this .establishes(7.35) for polynomial f.To extend (7.35) to rational f,inviewof partial fraction expansions, one need only consider the form (n m)-" (n > 0). One may define
+
(n+ m)-l = p"n"p";
(7.39)
then
(n+ m>(n+ m)-'u(x) = (n+ m)p-"n-' pmu(x) = p"nn -1 p m u(x) = u(x), .(n+ m)-l(n + m)u(x) = p-mn-lpyn + m)u(x) = p -m n-1 xprnu(x) = u(x). (7.40)
Thus the defini.tion of(7.39) preserves the commutativity of (n+ m)",(n+ my for any integral n,p and (7.35) is established for rational f. For the .specialcase u(x) 1, the operand will not be shown explicitly. One also has f(X)Prn
Since
m w .
=
(7.41)
Linear Equations with Polynomial Coefficients
+
f(n)pm = p"lf(m) +f'(m)n
207
e.]
(7.42)
= PMf(m),
(7.41) follows. A formula of some usein connection with monomial equations in n to be considered next is k
dk)u(X)= X'") A U@),
(7.43)
-1
that is, ~ (-n1)
*
9
k
(n- k + ~)u(x)= x'(x' - 1)
* *
(X'
- k + 1) A u(x). -1
(7.44)
From
n -j = pinp-j ,
(7.45)
one has (n- k + l)(n - k + 2). n = pk-1np-k+1pk-2np-k+2 . n I
-1
= Pk(P
4
I
k *
(7.46) (7.47)
Also, p-lnu(X) = / Y " [ X ' U ( X )
- dU(X - l)]
= E A u(x); -1
(7.48) (7.49)
hence, (p-'~= ) ~Ek
4k
(7.50)
-
and k
k
-1
P (P n) = x
! ( k ) E - k E k c\. k -1
k - x@)A -
(7.51) (7.52)
-1
(7.43) now follows. In view of (7.43), the typical monomial equation in n, namely
f ( n M 4=m , may be considered to have the form a,$") An U + ~,-~x'(n-I) An-1 u + " ' + a o u = g . -1
-1
(7.53) (7.54)
Chapter 7
208 The homogeneous equation
(7.55)
f(n)u = 0 has the solution u(x) = pk iff(k) = 0, since
(7.56)
f(n)pk= f ( k ) p k = 0. 2
Example: Lu = X’(’) A u - 2x’ A u -1
-1
+ 2u = 0. Thus
L=lt(7t-l)-227t+2=~~-337~+2
(7.57)
with roots k = 1, 2; hence the solution is
u(x) = p1(x)x’+p&)x’(x’.-
(7.58)
1)
with pl(x),p 2 ( x )arbitrary periodics. If the root k = a has multiplicity U, then (k - a)”is a factor o f f ( k ) ;hence aSf(k)/akPJk,a= 0 for S = 0, - , v - 1. Thus one may consider a
e
f(n>u(x>=f(k)Pk
(7.59)
and (7.60) Since, by the Leibniz rule, (7.61) it follo,ws that a‘/ale‘pkIk=a, j = 0,
I
Example: Lu = x’(2) A u - 5x’ A u -1
-1
I
, v - 1 are solutions o f (7.55), that is,
+ 9u = 0. One has
L = n(n - 1) - 5n + 9 = (n- 3)2.
(7.63)
Since (7.64) the complete solution is U(X) = X ’ ( ~ ) ( P ~ ( X+p2(x)+(x’ ) - 2)).
(7.65)
209
Polynomial Coefficients Equations with Linear
For the inhomogeneous equation (7.53), a convenient method of solution is to expand g(x) into a Newton series of the form (7.20) or into a series of inverse factorials of the form (7.21). The interpretation of each term of the formf(n>"pk*' is thenf(k fs)pkfs from (7.41) provided k fS is not a zero o f f (4. Example: (2- 3n + 2)u(x) = (x,+l/(x'+2), One may write (x2 - 3n
+ 2)u(x) = p-2,
(7.66)
hence a particular solution is 1 p-2 - $ 2 u(x) = - 1212 n2 - 3 n + 2
1 =-
1
(x' + l ) ( i + 2)
(7.67)
This plus the solution (7.58) of the homogeneous equation yields the complete solution of the example. Whenf(k fS) = 0 for some k, S, the interpretation off(n)"pk*S may be carried out by use of (7.39). If k f S is a multiple root, one may use the inductive extension of (7.39), i.e.,
(n+ m)-" = p"n"'pm.
(7.68)
Example: (n- 2)(n + l)u(x) = e",
a!
< In2. From the Newton expansion
m
(7.69) valid for a! < In 2, one has
c Oo
(e"
-
= s=O s ! ( n
1
(7.70)
- 2)(n + 1) PS;
thus (7.71) The exceptional term yields (e"
2
(e"
-
- (e*
-
1
(n- 2)(n + 1) P = -
6
6
P ;
1
(7.72) x'(x' -,l)+@' - 1).
Chapter 7
210
Another solution of this example may be obtained from 1
-11
11
(7.73)
” -”
(~-2)(~+1)-3n-2
3n+1’
Since
one has, for m = -2, 1 and g(x) = ea’, 1 u(x) - - x’(x’ -3
X’-1
S
1
ea‘
--
t2t+
1
At
(7.75)
”
3x’+1
4. GENERAL OPERATIONAL SOLUTION The solution of the general differenceequation with polynomial coefficients is effected by use ofboth the IC and p operators. For the following, it will be convenient to define the operator L by Lu = ao(x)u(x)
+ al(x)u(x - 1) + - + a,(x)u(x - m)
(7.76)
with aj(x) (0 5 i 5 n), as previously, polynomials. The complete equation Lu(x),= g(x)
(7.77)
may always be put into the form [b&)
+ bl (X)P + + b,(x)p“lu(x) = h(x) (7.78) * * *
by multiplication of (7.77) by quent use of
X’(,)
= x’(x’ - 1)
--
(x’ - n + 1) and subse-
pu(x) = x’u(x - l), p2u(x) = x’(x’ - l)u(x - 2), . . .
(7.79)
Polynomial Coefficients
Equations withLinear
211
An alternative procedure, which, however, leads to an equation for a different dependent variable, is the substitution
(7.80) From (7.16)and (7.32), one observes that
(n+ p + r)u = xu
(7.81)
and, hence, n+p+r=x.
(7.82) Thus x may be replaced by n + p + r in the polynomials &(x) (0 5 i 5 n) and (7.78) may be rewritten in the form
h(4+h(rc>P + ' ' ' +f,(4p"Iu(x)
=W
.
(7.83) According to Boole and Milne-Thompson, this will be called the canonical form of (7.77).The index m is calledthe order of the operator. When m = 0, one has the monomial equation (7.53)already discussed. If the equation is given in the form ao(x)
A u(x) + al(x)*. Minimize S, subject to u(0) = 1 , u(n) = 0. 5. Obtain the complete solution of ( x - l)u(x+ 2) - (3x - 2)u(x + 1 ) + 2xu(x) = 0
+
+
+
given one solution is u(x) = x.. 6 . Solve xu(x) - ( x l)u(x- 1 ) = 2. 7. Obtain the complete solution of
+
u(x
+ 2) - a ( 2 + l)u(x + 1 ) + d + ' U ( X )
= 0.
8. Obtain a particular solution
+ 3) + aXu(x+ 2) + aZXu(x+ 1 ) + a3xu(x)= aix2. 9. Solve: u(x + 3) - 7u(x + 2) + 16u(x + 1 ) - 12u(x) = l / x . u(x
10. Using the Laplace transformation, solve
a
+
+ +
--u(w, x ) = -x2u(w, x ) ( x l)'u(w, x 1 ) aw for ii(s, x). 1 1. Solve u(x> 7xu(x - I) 1Ox(x - l)u(x - 2) = X'. 12. Solve u(x) - 3xu(x - 1) + 9x(x - l)u(x - 2) = ax. 13. Solve (x2 -. 4)u(x)- (2x2- x)u(x - 1 ) x(x - l)u(x - 2) = 0. 14. Obtain a perturbation expansion in.&to two terms of the solution to u(x 1 ) - ( x Q ) U ( X ) = 0.
+
+
+
+
+
15. Obtain a singular perturbation expansion in E to two terms of the solution regular in E about the origin of EU(X
+ 2) - U ( X + 1 ) + 2 U ( X ) = 0.
References
1. RP Boas Jr. Entire Functions. New York: Academic Press, 1954. 2. NI Achiezer. Theory of Approximation. New York: Frederick Ungar, 1956. 3. C Truesdell. A Unified Theory of Special Functions. Annals of Mathematics Studies No. 18. Princeton, NJ: Princeton University Press, 1948. 4. W Gruber, H Knapp. Contributions to the Method of Lie Series, Mannheim: Hochschultaschenbiicher-Verlag,1967. 5 . EJWatson. Laplace Transformsand Applications. Van Nostrand Reinhold,
1924. 6. B Sengupta, DL Jagerman. A conditional response time of the M/M/l processor-sharing queue. BSTJ 64:409421, 1985. 7. B Sengupta. A perturbation method for solving some queues with processor sharing discipline. J Appl Prob 26:209-214, 1989. 8. LM Milne-Thomson. The Calcululs of Finite Differences. London: Macmillan, 1933.
9. C Jordan. Calculus of Finite Differences. New York: Chelsea, 1947. 10. G Doetsch. Theorie und Anwendung der Laplace-Transformation. New York: Dover, 1943.
237
References 11. DV Widder. The Laplace Transform. Princeton, NJ: Princeton University Press, 1941. 12. NTJ Bailey. The Elements of Stochastic Processes. New York: Wiley, 1964. 13. TE Harris. Theory of Branching Processes. New York: Springer, 1963. 14. MS Bartlett. An Introduction to Stochastic Processes. Cambridge: University Press, 1956. 15. T Fort. Finite Differences and Difference Equationsin Oxford: Oxford University Press, 1948.
the Real Domain.
26. R Bellman. Dynamic Programming. Princeton, NJ: Princeton University Press, 1957. 17. NE Norlund. Vorlesungen iiber Differencenrechnung. Berlin: Springer Verlag, 1924. 18. Bateman Project. Higher Transcendental Functions. New York: McGraw-Hill, 1953. 19. Hardy, Littlewood, Polya. Inequalities. Cambridge: University Press, 1959. 20. J F Steffensen. Interpolation. New York: Chelsea, 1950. 21. K Knopp. Theory and Application of Infinite Series. London: Blackie & Son, 1946. 22. VI Krylov. Approximate Calculation of Integrals. New York: Macmillan, 1962. 23. NK Bary. A Treatise on Trigonometric Series. New York: Pergamon, 1964. 24. W Magnus, F Oberhettinger. Formulas and Theorems fortheFunctions Mathematical Physics. New York: Chelsea, 1954.
of
25. TJI’A Bromwich. An Introduction to the Theory of Infinite Series, London: Macmillan, 1908. 26. RB Cooper.Introduction 1981.
to Queueing Theory. New York: North Holland,
27. L Kosten. Stochastic Theory of Service Systems. New York: Pergamon, 1973. 28. DL Jagerman. Methods in traffic calculations. AT&T Bell Lab Tech J 63(7): 1984. 29. DL Jagerman. “Some Properties of the Erlang Loss Function,” BSTJ53(3): 1974. 30. L Taka’cs. Introduction to the Theory of Queues. Annals of Mathematics Studies No. 18. Princeton, NJ: Princeton University Press, 1948.
References
239
31. HAkimaru,T Nishimura. The derivatives of Erlang's Bformula. RevElec Commun Lab ll(9-10): 1963. 32. R F Rey. Engineering and Operating in the Bell System. 2nd ed. Murray Hill, NJ: ATLT Bell Laboratories, 1983. 33. DR Cox. Renewal Theory. New York: Methuen, 1962. 34. RE Mickens. Difference Equations. Van Nostrand Reinhold, 1987. 35. S Karlin and HM Taylor. A First Course in Stochastic Processes. New York: Academic Press, 1975. 36. PL Butzer, H Berens. Semi-Groups of Operatorsand York: Springer-Verlag, 1967.
Approximation. New
37. G H Hardy. Divergent Series. Oxford: Clarendon Press, 1949. 38. R Askey. Ramanujan's Extensions of the Gamma and Beta Functions. 39. ND de Bruijn. Asymptotic Methods in Analysis. Amsterdam: North Holland, 1958. 40. TL Saaty. Modern Nonlinear Equations. New York: Dover, 1981. 41. DL Jagerman. An inversion technique for the Laplace transform with application to approximation. BSTJ 57(3): 1978. 42. DL Jagerman. An inversion technique for the Laplace transform. BSTJ 61(8): 1982. 43. G Boole. A Treatise on the Calculus of Finite Differences. Stechert, 1931. 44. J Riordan. Stochastic Service Systems. New York: Wiley, 1962. 45. L Kleinrock. Queuing Systems. Vol 2. Computer Applications. New York: Wiley Interscience, 1976. 46. K Jamsa. Java Now! Las Vegas, NV: Jamsa Press, 1996.
,
,
,
,.,,
,
. , ,, ,,,
,
_... ,
, , ,
,,".
,
... . . _
..._... ..
,
... , . . . ..
.
.,
.. ...
.... "..... ....
II
,... ..
. .
,
.
.
.. .. "...
Index
A Alternate representation of sum, 89 Approximation to generators, 161 Approximation method, 176 approximate inversion, 176 approximation to P(t,O), 177 Approximation to solution, 114 error estimate, 114 B Bandlimited functions, 105 samples on half line, 105 Bernoulli’s polynomials, 50 Bernoulli numbers, 5 1 complementary argument theorem, 54 generating function, 54 generating function for Bernoulli numbers, 55 multiplication theorem, 53 problem of Bernoulli, 51
[Bernoulli polynomials] relation to zeta function, 55 table of Bernoulli numbers, 53 table of Bernoulli polynomials, 52 Beta function, 7 Birth-death model, 137 solution, 138 Bound for remainder, 83 Branching process approximation, 145 critical case, 146 nonnegative variable, 146 subcritical case, 146 supercritical case, 145 Branching processes, 1.36 generator for population, 137 C Casorati’s determinant, 202 Casorati’s theorem, 26 construction of a difference equation, 29
242
Index
[Equation of nth order] Class of linear transformations, 89 associated function, 90 examples, 158 Bore1 transform, 90 Equations reducible to constant exponential order of sequence, 89 coefficients, 188 maximum modulus, 90 Erlang loss function, 161 vector space, 90 Error bound for transformation, Coefficients A,(@), 127 101 Complete equation, 222 derived error bound, 102 Constant coefficients, 172 error transformation, 101 characteristic equation, 173 Euler equation, 25 homogeneous equation, 173 Euler summability, 159 multiple roots, 173 application of summability, simple roots, 173 161 Convergence of infinite products, 1 13 circle of summability, 160 Convolution integral representation, consistency,160 101 regularity, 160 derived error bound, 102 Euler-Maclaurin expansion, 75 error estimate theorem, 102 periodic Bernoullian functions, Critical points, 143 75 classification,144 usual form of expansion, 76 Exact expansion for P(t,O), 177 D Exceptional cases, 217 multiple roots, 217 Depression of order, 201 multiple roots - Newton series, Derivative, 129 219 approximate mean work in Existence of principal’sum, 77 M/G/l, 131 Expansion for g(z), 142 approximation for Erlang loss approximation to g(z), 143 function, 130 Expansions and functional approximation to u’(x)/u(x),130 equations, 96 Differential-difference equations, differential-difference equation, 122 97 examples of F-equation, 124 first order difference equation, 98 method of Truesdell, 122 Newton’s expansion, 96 reduction procedure, 123 Differentiation, 39 Differentiation formula, 81 asymptotic properties, 97. 82 mth derivative of sum, 82 E
Equation of nth order, 158
F Factorial function, 4 Factorial series, 7 first kind, 7 second kind, 7 uniqueness, 9
Index
243
Finite source model, 195 Functional equations, 132 G Gauss-Laguerre quadrature table, 6 1 Gauss-Legendre quadrature table, 161 General operational solution, 210 inverse factorial series, 212 reduction to canonical form, 211 use of R,213 Genesis, 21 Clairault, 22 geometric distribution, 22 Poisson distribution, 22 Riccati, 22 two parameter family, 23 GI/M/1 queue, 125 Gregory-Laplace quadrature, 64 H Haldane’s method, 149 expansion for l/g(z), 150 Heymann’s theorem, 29 Homogeneous equation of order n, 25 congruent points, 26 fundamental system, 26 general form of solution, 29 Homogeneous form, 44 bounds for eigenvalue and sum, 46 commutability, 45 convexity of eigenvalue, 47 derivative, 44 difference, 44 eigenfunction of homogeneous sum, 45 extension of summability, 45
[Homogeneous form] representation of Laplace transforms, 45 Homogeneous sum exponential weight, 100 1
Identity in z, 207 Indicia1 equation, Inverse factorial series, 213 Inhomogeneous equation, 179, 114 Boole’s operational method, 179 concomitant boundary condition, 115 construction of solution, 114 method of Broggi, 182 partial fraction method, 182 specification of u(O), 116 useful form of solution, 115 Inhomogeneous form, 197 Integral equations, 102 inhomogeneous, 104 Invariant function, 157 Inverse of z + m,206 1
Lagrange’s method, 190 Lagrange’s variation of parameters, 224 Landau, 7 associated series, 7 convergence, 7 Laplace numbers generating function, 63 Laplace, 12 abscissa, 13 bilateral, 13 convergence, 12 convolution, 13 operational properties, 14
Index
244
[Laplace] transform, 12 Laplace numbers, 62 Laplace numbers table, 63 Laplace’s method, 183 Laplace’s observation, 193 Boole’s example, 194 LCFS M/M/C queue with reneging, 226 final solution, 231 formulation and solution, 228 method for binomial equations, 229 perturbation solution for small alpha, 231 second solution, 229 test customer, 226 Level crossings of work, 120 Linear homogeneous equation, 110 gamma function, 111 infinite product solution, 113 multiplication formula, 111 solution in terms of gamma function, 112 Linear transformation, 91 class A, 91 eigenfunction, 93 generating function, 91 inverse, 94 sum and product, 93 Lower bound theorem, 49 M M/G/l queue with reneging, 121 difference equation, 121 M/M/1 case, 122 Volterra integral equation, 121 M/M/l delay model, 24 differential-difference equation, 24 semigroup type, 24
M/M/l processor sharing queue, 234 singular perturbation solution, 235 tagged customers, 234 M/M/ 1 queue transient solution, 174 exact inversion, 176 Laplace transform of solution, 175 M/M/l queue with feedback, 132 M/M/C Erlang blocking model, 24 Mellin, 14 convolution, 14,15 inverse factorial series, 14 operational properties, 14,15 product of series, 16 Milne-Thomson, 30 asymptotic criterion, 30 Monomial equations in 5t, 207 multiple roots, 208 Monomial equations in p, 204 Erlang loss function, 204 use of factorial series, 204 N Newton’s expansion, 2 defined, 2 uniqueness, 8 Nonlinear difference equation, 133 generating function g(zlh), 133 ordinary differential equation, 133 partial differential equation, 134 Norlund, 8 uniform convergence, 8 Norlund principal sum, 75 Norlund sum for complex arguments, 99 complementary argument formula, 100
Index
245
[Norlund sum for complex arguments] complementary argument theorem, 100 expansion for, 99 integral representation, 99 Norlund’s expansion, 56 asymptotic properties, 57 error estimate, 57 homogeneous sum, 59 representation of sum, 59 Null function, 13 Numerical differentiation, 60 0
Operational formulae, 17 shift formula, 17 Operational method of Boole, 179 Operators, 1 Operator n, 205 Operators n and p, 203 P Partial difference equations, 189 Bernoulli trials, 190 Pascal’s triangle, 189 Perturbation solution, 147 Norlund sum expression, 147 perturbation solution, 148 Principal solution, 33 limits and span, 33 Norlund definition, 33 sum, 33 Properties of sum, 34 limiting form, 36 linear change of variable, 34 multiplication theorem, 35 span integral, 35 sum from primitive, 36 Psi-function, 10
Q Quadrature formula for sum, 62 Quadrature of homogeneous sum, 65 coefficients for exponential weight, 66 coefficients for linear weight, 68 Queueing model M/M/n, 118 asymptotic expansion, 120 differences of l/B(O,a), 120 Erlang loss function, 118 expansion for l/B(x,a), 119 extension to continuous argument, 1 19 Fortet integral, 120 R Reduction of order, 21 3 Repeated summation, 40 Representation theorem, 90 Riccati form, 135 inifinitesimal generators, 136 solution, 135 Roots differing by an integer, 220 5
Separation of variables, 191 Sequence of transformations, 111 convergence of generating functions, 95 convergence theorem, 95 Shift formula, 205 Simultaneous first-order equations, 154 equivalent differential equations, 154 equivalent partial differential equations, 154 limiting form, 157
Index
246
[Simultaneous first-order equations] Newton’s expansion, 156 partial differences, 156 U-operator method, 155 Solution in Newton series, 215 descending factorials, 216 example - integral form of solution, 2 16 indicia1 equation, 21 5 Solution of functional equation, 152 example, 153 example of M/M/l queue with feedback, 152 solution of special case, 153 Stirling equation, 192 Stirling numbers, 4 first kind, 5 second kind, 5 Sum of Laplace transforms, 41 representation theorem, 41 Summability, 39 Summation by parts, 38 Summation of series, 11 from sum of function, 37 T Taylor’s series, 3 Time-homogeneous model, 25
Trigonometric expansions, 84 coefficients for, 85 example of, 86 +-function example, 87 removal of discontinuity, 87 sum of exponential, 86
U Uniqueness theorem, 92 U-operator solution, 138 accuracy dependent on h, 141 Hille’s representation, 140 infinitesimal generator, 140 invariant function, 141 modification for good accuracy, 141 Newton’s series expansion, 139 reduction to Lie-Grobner operator, 139 relation to semigroups, 140 Use of Laplace transform, 184 example, 186 principal solution, 184
v Volterra integral equation, 65
