Handbook of special functions: derivatives, integrals, series and other formulas

H A N D B O O K O F Special Functions Derivatives, Integrals, Series and Other Formulas H A N D B O O K O F Speci...

Author: Yury A. Brychkov

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H A N D B O O K

O F

Special Functions Derivatives, Integrals, Series and Other Formulas

H A N D B O O K

O F

Special Functions Derivatives, Integrals, Series and Other Formulas

Yury A. Brychkov Computing Center of the Russian Academy of Sciences Moscow, Russia

Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2008 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-58488-956-4 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

Contents

The Derivatives # " # $!%" & ' & !! ( ( )* ' !!"!! '' **, + + * *, " " & +**,!"! & ( # +**,!"! ( ' +**,!"! . & +**,!"! . . ! ! ( +* *, " . !!"!! **, -- + " " , * * + ! ! / , " * * / + ! ! 0 +* *, " # 1

1.1.

Elementary Functions

1.2.

The Hurwitz Zeta Function ζ (ν, z )

1.3.

The Exponential Integral Ei (z )

1.4.

The Sine si (z ) and Cosine ci (z ) Integrals

1.5.

The Error Functions erf (z ) and erfc (z )

1.6.

The Fresnel Integrals S (z ) and C (z )

1.7.

The Generalized Fresnel Integrals S (z, ν ) and C (z, ν )

1.8.

The Incomplete Gamma Functions γ (ν, z ) and Γ(ν, z )

1.9.

The Parabolic Cylinder Function Dν (z )

!!"!! ****, .. + + , " 0 ! ! , " * * + +**,!"!0 !!"!! ****, + + , " 0 !!"!! **, + 0 , " * * + !!"!! ****, ## + + , " 0 !!"!! ****, '' + + , " 0 !!"!! ****, && + + , " 0 !!"!! ****, (( + 0 + , " - +**,!"! / +**,!"! . +**,!"! !!"!! ****, + + , " " !!"!! ****, + + , " " !!"! ****, + + , "" 1

1.10.

The Bessel Function Jν (z )

1.11.

The Bessel Function Yν (z )

1.12.

The Hankel Functions Hν(1) (z ) and Hν(2) (z )

1.13.

The Modified Bessel Function Iν (z )

1.14.

The Macdonald Function Kν (z )

1.15.

The Struve Functions Hν (z ) and Lν (z )

1.16.

The Anger Jν (z ) and Weber Eν (z ) Functions

1.17.

The Kelvin Functions berν (z ), beiν (z ), kerν (z ) and keiν (z )

1.18.

The Legendre Polynomials Pn (z )

1.19.

The Chebyshev Polynomials Tn (z ) and Un (z )

1.20.

The Hermite Polynomials Hn (z )

1.21.

The Laguerre Polynomials Lλn (z )

1.22.

The Gegenbauer Polynomials Cnλ (z )

1.23.

The Jacobi Polynomials Pn(ρ,σ ) (z )

' ' / / ' & &. # # # #' #' ##( #-#' '& '& '/ '/ & & & &&# &# &# && && && &/

/ & ! ! # +* *, " &./ (. ! ! ' , " * * ' + +**,!"" (( ( ! ! ' **, " && + ( ! + * *, " " ( (& ! ! , " * * (( + (& +**,!"" -. (( - +**,!"! -. -. ! ! / -. **, " / + ! + * *, " " --' !!"! --&& **, .. + + * *, "" ( /' /' /' !!0 0

! ** 0 /' 0"% /' # !! % 0 //&& ' ! !% !*" & $"% /( ( !"% /( - !"%! //. !"% / $%" // " .. !! . 0

# . !## # . 1

1.24.

The Complete Elliptic Integrals K (z ), E (z ) and D (z )

1.25.

The Legendre Function Pνµ (z )

1.26.

The Kummer Confluent Hypergeometric Function

1 F 1 (a ; b ; z )

1.27.

The Tricomi Confluent Hypergeometric Function Ψ(a; b; z )

1.28.

The Whittaker Functions Mµ,ν (z ) and W µ,ν (z )

1.29.

The Gauss Hypergeometric Function 2 F1 (a, b; c; z )

1.30.

The Generalized Hypergeometric Function p Fq ((ap ); (bq ); z )

Limits

2.1.

Special Functions

H

Indefinite Integrals

3.1.


3.2.

Special Functions

L

!!*00 .' # % .' ./ " ## ! & ### $%" ( ! ! ' '' ##& )* 0 ( ## ))0 ( / ( 0 ! ! -. ### ))0* -. -- 0 ## ))0 - 00!! - ### )) 0 * 0 - ' # )" --# 0 0! --'## #### )) 0 " ! -& ! ##### )) 0 0* ( ##' ) ---( 0 0 " ##& ) -/

00 -/ ##'' )) 0! ! /. ' / ##'# )) 0 / ' ' 0 / ##'& )) 0!! / / 00! / ##&& )) 0" / / #& ) 1 H

L

Definite Integrals

4.1.


4.2.

The Dilogarithm Li2 (z )

4.3.

The Sine Si (z ) and Cosine ci (z ) Integrals

4.4.

The Error Functions erf (z ), erfi (z ) and erfc (z )

4.5.

The Fresnel Integrals S (z ) and C (z )

4.6.

The Incomplete Gamma Function γ (ν, z )

0!! /' #&#' ) 0 0 /' #& ) #&& ) " /' /& 0 /& #( ) 00!" //( #( ) ! ! //( #( ) 0 // #(#' ) 0 0 .. * #( ) #(& )"0 . #(( ) . 0 .## #-- ) .# # ) 0 .' 0 .' #// ) 0!" .& #/ ) 00! ! .#/ ) 0 . #/#' ) 0* #/ ) " 0 #/& )" # ( ) & 0

. 0 & #. ) # ) " & ( 0 ( # ) 0! / %" # ) 0 / 0! ! . # ) 0 ##' ) * # )

0 # ) 0 0 # )" 0 ## ) 00! ! ## ) & ## )

4.7.


4.8.

The Bessel Function Yν (z )

4.9.


4.10.


4.11.

The Struve Functions Hν (z ) and Lν (z ) H L H H L H L H L

4.12.

The Kelvin Functions berν (z ), beiν (z ), kerν (z ) and keiν (z )

4.13.

The Airy Functions Ai (z ) and Bi (z )

4.14.


0 0 #####' )) " # 0 ' 0 ##' )) # #' )0" #& 00 # ##&& )) # # 0

##&&# ))"0 #& #& 00! #( ##(( )) -/ " #( # 0 0 '# ##((# )) 0

' ##((& ))"0 ' ' '# 0 '# #-- ) 0 '' #- ) 0 '& #- ) ## ) "0 '& '( 0 '( #// ) 0 '( #/ ) # ) "0 & & ! !0 & #.. ) !0 & #. ) 0! & #. ) # # ) "0 &#' !0! 0 &&' ## )) " %" &( 0 ! ! ### )) 0* &( #' ) 0 (#

4.15.

The Chebyshev Polynomials Tn (z )

4.16.

The Chebyshev Polynomials Un (z )

4.17.


4.18.


4.19.


4.20.


4.21.

The Complete Elliptic Integral K (z ) K K K K K

0 (# #& ) 0 (' #-( ) 0 (' #/ ) 0 (' # . ) 0 (& #) 0 (( #) 0 0 (#) # ) " (-/ 0 ((/ # ) !"!%"0 # ) 0 ! ! - & # ) 0* ##' ) 0 -/ # ) 0 / #& ) 0 / #-( ) 0 / #/ ) 0 / # . ) 0 / # # #) 0 / & #) 0 / / #)"00 /& //& #)"0 # #) .. 0% .. # ) . # ) "0 0 . 0 . ## ) 0 .# ## ) 0!! .& ## ) 0 .& ###' ) # # ) "0 .& ./ ./ ' ./ ' " 0 ' K K K K K K

H

L

K

K

4.22.

The Complete Elliptic Integral E (z ) E E E E E E E E E E E

H

L

E

E E

4.23.

4.24.

K

The Complete Elliptic Integral D (z ) D D K


K

Finite Sums

5.1.

E

The Psi Function ψ (z )

E

'' 0 " ' # ' 0 # ' " # ' ''# 0 0 '' "0 '## " ( ''''

- 0

" '' "0 / / '& 0 / . ''( 0 . ' 0 " '(( " ' '(# & ''(' 0 (& " / ''-- 0 / "

''-- 0 '-#' ' " '-& ' ( ''// 0 0 " #.( '/ # ''//#' " ## '/& "0 ## ## '. ##

5.2.

The Incomplete Gamma Functions γ (ν, z ) and Γ(ν, z )

5.3.


5.4.


5.5.


5.6.

The Struve Functions Hν (z ) and Lν (z ) H L

5.7.


5.8.

The Chebyshev Polynomials Tn (z ) and Un (z )

5.9.


5.10.


#& '. #'. 0" '' '.# '( '.' "0 '/ '.& 0" & '.( & '.- "0 & & ' ' &' ' &' 0" ( '# '' "0 ( - '& '( 0" -'- "0 /. / ' !! / ' / ' !0" #. '# "0 ! #.' '' ! #.' '& ! 0" #.'( "0 ! #. ' # # # ''# # 00" '## # " # # '''' # 0 " ## # '& # ##

5.11.


5.12.


5.13.

The Legendre Function Pνµ (z )

5.14.

The Kummer Confluent Hypergeometric Function

1 F 1 (a ; b ; z )

5.15.

The Tricomi Confluent Hypergeometric Function Ψ(a; b; z )

5.16.

The Gauss Hypergeometric Function 2 F1 (a, b; c; z )

'& 00" ' '& " # #('(

0 # # # '( '(

" # 0 '(# " ## '(' "0 ' '(& ##' #& ''-- # & ! " % #( #/ / # / && # ! / " # ! . # &&# %" # # & 0 # 0 && ##( ##( " &# ##'/ # & #''

# ##'' &# #0 #' &# #' &&### "0# # #'

#

#00 ##' &&'' #'' # #

0

' # # # #'&' 0 & # "

#0 # #'/ #'/

&& "0 #'/. && #& 1

5.17.


5.18.

Multiple Sums

Infinite Series

6.1.


6.2.

The Psi Function ψ (z )

6.3.

The Hurwitz Zeta Function ζ (s, z )

6.4.

The Sine Si (z ) and Cosine ci (z ) Integrals

6.5.

The Fresnel Integrals S (x) and C (x)

6.6.

The Incomplete Gamma Function γ (ν, z )

. #& 0 &( % #&. #& &-- #& , &- ! #&' &- #& &-#' 0 #&& &- # #&( &-& #0 #&( &--( "0 # #(.. &-/ "0 #0 #( &- . #0 # #( &- # #0 # #( &- #( &- #( #(# &- & # #('' #(' / &/ 0 #( &/ "0 #(& &/ #(( -. 0 &/#' " # & #-. - # 0 #- &.. # #- &.

0 #- &. ## &.#' # 0 # #0 # #--# &. #0 # #-'# &.& "0 # #-' # &( -& # 0 #--& & #& & 0 " & #- #-&# -/ # 0 #-/ & #-/ & 1

6.7.

The Parabolic Cylinder Function Dν (z )

6.8.

The Bessel Functions Jν (z ) and Yν (z )

6.9.


6.10.

The Struve Functions Hν (z ) and Lν (z ) H L H H H H H H

6.11.


6.12.

The Chebyshev Polynomials Tk (z ) and Uk (z )

0 #/. & #/. &# / # 0 / && # 0 " #/

0 #/ & #//# " &#' 0

" ## & /' # #/' &# #/& 0" &# "0 #/( &# "0 # #/( &##' #/( &# 0" #/ &#& 0 &#( " #/// # #// &'' '.. 0" &' 0 " '. &' '. &'#' & 0" '.# '.' ! && ! 0 '.' '.& " && 0 && "! ! '. ( '. &&#' ! 0 '.( && " ( '.

'.&&(( ' 0

0 ' " &&((# "0 '#( # '( &&((&' '

0" ' &(-( " 0 '& &( 1

6.13.

Hermite Polynomials Hn (z )

6.14.


6.15.


6.16.


6.17.


' ( ' '( ( ' ( '( ! " (( !" 0 '-( %0! ! ' ( !!" 0 '/ 0 ((#' ! '# * ! 0 ''#' ((&( ! % 0 # !

* ((/ !0"% 0 ''#& / # ! ! ! 0 %" *" '''#/ % ((. !$ % % '' " ( !! "%! '' ( ! ''' "% ((#' !"% ! % '' ! " " ''/ ((&( ! " " '&. ! 0 (- '& ''& '& -- !!!!%" & '-/ - !!%" &. --#' !!!!%" & %" &-/ --& !!!!%" %" & & --( !!%" & %"

!%" & --/. !! ! %" & -- !!!!%" &' &/ - !!%" &/ %" --#' !!!!%"

& & %" -& !!%" & 1 The Connection Formulas

7.1.


7.2.

Special Functions

H

J

L

E

K E

Representations of Hypergeometric Functions and of the Meijer G Function

8.1.

The Hypergeometric Functions

---( !!!!%" & & %" --/. !!!!%" & &'# -- !!!!%" & %" && %" - ! 0! %" &( & & -- &/ ' -- % &&' " &'' &&/ 0 &(/ ))00 % &( (a p ) (b q )

8.2.

The Meijer Function Gm,n z p,q

8.3.

Representation in Terms of Hypergeometric Functions

1

Preface

$!% &'(!#%)!"* +" ,&")-.!%&!"-)/" * !"#! ,& )!-$,.))!%?7A@CB

U4. & - U ')( $ XW4. ' + & + 0& ¤¤4¤ $ $ L V V L

U4 J 1 XW4 J (_BB B?7R@CB 13.

0 f / ' & f a 1

I+ 1A- . & -21 $9% '6 ')5 ( $ $ ! ! 43fI - - & - -21 f / = '*>?7R@CB m ! 20. ' & f a 1 - + a 1 . & -21 $9% '6')5 ( $ $ ! ! 3PfI - - a f / &- 1

= '*>?7R@CB m ! 21. '%& f a 1 - + a 1 . & -21 $&% '6' 5 ( $ $ ! ! 43fI - - -;&-21 0f / = '*>?7R@CB 19.

1.1.5. Trigonometric functions 1.

0fI& f

F fI

' ¤ G
7R@CB ¢ 43 3 1 &-21 3 £! 4f;/ = ? S 25. 0f;/ ' & S £) a ¤ £) 43 ¢ ( & '% & m '#" 3 8 J $ L 3 1 &-21 4 3 4 f / 22.

26. 27. 28.

29.

30.

31.

m 2! m a . H 3 ¢ / *F G 1 - + 1 a a 1 0f / ¤ ! m m a . ED H / *F G 1 - + 1 a -;&-21 f;/ ¤ a / ?F ( G 1 7 ( &B BB ( 5 9BB B 5 #$ & 1 1 1 1 Y a N a

& 1 & ( &B BB & ( & 5 &B B B & 5 ! ( 7

' %

% ] ( 7X ! ' ' %

¢ a D H 3 1 ' & '%& 7 V( &B BB V( &B B B ( 7 #$ ! Y a N a

& 7 ( &B BB #7( #7?7A@CB

38.

f / = '8>?7A@CB

39.

0f / = '8>?7A@ B

H a 1;D H a D H a & D H D H a 1 D H

41.

42.

43.

44.

45.

RED

40.

R R R R R

3#

3#

643 ¢ £ a 1

3V

3 ¢ a 1 £ a 1

3V

¤

¤

¤

& ¤

& ¤

& ¤

R a & D

47.

R a D

46.

48.

49.

50.

51.

H 643 ¢ £ a 1

:

H 3 ¢ a 1 £ a 1

¤

a 1

& ¤

m . 0f;/ ¢ / 8 G F I +

A 1 f;/ Q ' 5 ! £ £ ?7R@CB

F f f GIH ) ' (87! % ) ' (:7X! % m f f9

-; j & -21 F 5 m G 3 m m 5

ED m (8?7R@CB = '8>?7A@CB

m 5 m ( ?7R@CB = '8>?7A@CB = '8>?7A@CB m 5 ¤ m (

m 5 ¤ m ( m 5 ¤ m ( = '8>?7A@CB = '8>?7A@CB ¢ 3hfI& ¤

m fI& 0 3A &-21 3 ¢ f ¢ h 3 f9

-; j &-21 F 7( m G ' >?7R@CB =* 3

2.

3.

4.

5.

6.

7.

0 fI& m 3 ¢ &-21 3 ¢ f ¢ h 3 f9

-; j & -21 F 7 ( m

fI& 0 7 43 ¢ £ 3 ¢ f9 a 1 ; ¢ f9 O

-;&-21

&-21 F 7?7R@CB G

= '*>?7R@CB 7

¤ 7?7R@CB = '*>?7R@CB = '*>?7R@CB f;/ ¤ 0f / ¤ 0f;/ ¤

14.

15.

16. 17. 18. 19. 20. 21.

22.

23.

24.

k a 1 ¢ h 3 f & a 1 k -21 ¢ h 3 f & -21 f;/ f9 f / ¤ a 1 ¢ 3hf9

& -21 -21 ¢ 3hf9

& &-21 f / £ 3 m ¢ 3hf9

& -;&-21 f;/ ¤ , ) k 1 ¢ 3hf9

& &-21 &-21 f;/ 0 ¡ = '8>?7A@CB

k &-21 ¢ f9

& P f;/ ¡ k -21 ¢ f9

& Pk &-21 f / £ F3 k a 1 ¢ f9

& Pk -21 f;/ £ k &-21 M ¢ f & &-21 f;/ F 7 G f -21

k a 1 k -21 ¢ f9

& &-21 F 7 G f9

7 G -;&-21

0f;/ ¤

f / ¤

¢ f & -;&-21

f;/ ¤

0f;/ ¢ 9f

& -;&-21

f;/ ¤

3 m , )

1.2.1. Derivatives with respect to the argument

2.

fI& EDM &-21 F

3PfI ; ?7A@CB

1.2. The Hurwitz Zeta Function ζ(ν, z) 1.

= '8>?7A@CB = '8>?7A@CB

k &-21 ¢ h 3 f9

& 1 ¢ h 3 f9 O& &-21 0f / ¡ k &-21 ¢ 3hf & &-21 f;/ ¡ k 1 ¢ f9

& &-21 f;/ ¡

2.

3. 4.

5.

6.

7 ' a $ £) 3 £ F '*G 7 - 1 3 $ ' " £! 3 £ ( 7X! " & -21 . ( 7! " ] $ (:7X! % " 3 ' ! ' + -21 F 'PG 3 O' ! " 1 7 Y & -21 ' F £) ' G ' " 43 £! ¢ = ' 7 1

F G a 3 ' ' £ 1A- 3 ¢ 3 £ ¢ - 1 C7( " ' ! 7 F G a C7( " ' !0' - 1 ( 7X! ' 7 ] ' " ' # ' " '#" + &-21 . F G 3 ' " ' (:7 3 £ ¢ C7 (* " ' ! 7 G a ' F

C7( - " ' ! 1 ( 7! " . 7 £ % ' & # ' " # ' " 3 ' " ' + &-21 F G 7( 3 '#" 3 £ ¢ C7( " ' !AC7?7A@ B

$! "!$ #

" & -21 m ]! ¢ GH Q 3 3 -21 - R $&% 3. EDM &-21 F 3 9 3 ¢ 3 ¢ -21 -R -; F mG 4. &-21 " & -21 ( 7X! "'& $&% f 43fI& f m ]! 5. 3fI& 0 m 6. EDM &-21 R F]3 GIH 43fI -;&-21 R F]3 m G 43 ¢ f &-21 -; & -21 3 ¢ F G m 3 ¢

-;&-21 43fI& ¤ 7. -21 - P 3PfI& a - R ED -21 R F3 m G9H H 43 ¢ 4 F]3 m G ¤ 8. EDM 1 f 3PfI& ¤ 9. V- 3PfI& m m 10. ED - R PED &-21 R F]3 GIHOH f -;&-21 F]3 G ¤

m

= '8>?7A@ B = '8>?7A@ B = '8>?7A@ B = '*>h7A@ B

1.4. The Sine si (z) and Cosine ci (z) Integrals 1.4.1. Derivatives with respect to the argument 1. 2.

3. 4.

5.

')(:7X! % -; 0fI& m EDM &-21 F GIH 7! % 43 ¢ ' (8 D ')(:7X! % -; 0fI&

&-;-2 1 3AfI& 3 V -

m G 3 #- R R -; &-21 F 3 -; 43 AfI& - &-21

&-;-2 1 AfI&

= '8>?7A@ B

m ;&- -2 1 F 9G H &-;-2 1 AfI&

= '*>?7R@CB = '8>?7A@ B

m EDM &-21 F G9H 4 3 ¢ ')O(8 7! % D R ;- F3 m G #- R ;- F m G9H &2- 1 &2- 1 & h7A@ B

= '*>h7A@ B

m - R & 1 2- 1 -; F IG H > 7R@CB = '*?

1.7. The Generalized Fresnel Integrals S(z, ν) and C(z, ν) 1.7.1. Derivatives with respect to the argument 1.

2.

3.

fI ; ') 0 3 (: 7X! % f# -; -; 3AfI& 3 - -; AfI& &-21 &-21 m D &-21 F GIH 43 ¢ &-21 ')(: 7! % f V - -21 D R -; F3 m G 3 - R ;- F m &-21 &2- 1 0fI ') (:7X! % 3 f# -;

&;- -2 1 3AfI&

,9N

-

&;--2 1 AfI&

= '*>?7R@CB G9H = '*>?7R@CB = '*>?7R@CB

4.

m EDM & -21 F G9H 7! % m 43 ¢ &-21 ')(: fV - 2- 1 D R -; F 3 G 3 & -21

m - R & -;-2 1 F 9G H ' >?7R@CB =*

1.8. The Incomplete Gamma Functions γ(ν, z) and Γ(ν, z) 1.8.1. Derivatives with respect to the argument

fI& 0 3 ¢ fV -; - & -;-2 1 fI& m 3 ¢ 3 ¢ f# - -21 -R -; F m 2. EDM &-21 F GIH &-21 ¢ < ¤ 43 - -; fI& 3. - fI& 0 m -21 F ?7A@ B

1.

11. 12. 13. 14. 15. 16. 17.

G

G ¤ ¤

m F G ¤

fI& 0 ; 43fI fI& ¤ m m ED &- R PED a -21 F GIHOH 3fI -;&-21 R F G ¤ -21 - ?7A@ B &-21 ¢ _ ¤ 3 - -; - fI& m GIH -21 F ?7R@CB F m G ¤

7 ' m ¢ ED &-21 F IG H J $ LF G F G &m m m . ED -21 F GIH F G - + -21 F G ¤ ( 7X! ' m m m ¤ . EDI+ &- a 1 F GIH F G &-21 -;&-21 F G 7 m m m . EDI+ &- -;&-21 F GIH F G &-21 -;&-21 F G ¤ m . EDI+ &-21 a 1 F GIH m ¤ 43 ¢ / 8F m G a 1 -21 a F m G G F 1

-;&-21

m ! '#" &-21 &-21 fI& 0 &-2 1 = '8>?7A@ B m

14.

N1

,9:

16. 17. 18. 19.

20. 21. 22. 23.

24. 25. 26.

27.

( 7! ' £ = '8>?7A@ B D -21 R &-21 F 9G H If &-21

- R m ! '#" &-21 fI& &-21 fI& 0 &-21 £ fI& ¤ m ! ']" &-21 0fI& &-21 fI& 0 &-21 £ fI& = '8>?7A@ B m ! '#" ¢ a &-21 fI& 1 -; fI& 0 3 1 &-21 £ fI& = '*>?7A@ B m ! '#" ¢ &-21 0fI& 1 -; fI& 0 3 &-21 £ fI& ¤ m ¤ m ( 7X! ' £ m D -21 &-21 F GIH fI &-21 - m m m ( 7X! ' £ ED -21 &-21 F GIH = '8>?7A@ B fI &-21 - ' m 0f;/ F G 43 ¢ J $ L a f;/ -; a 0f;/ ¤ m ']" . £ &-21 & -21 0f;/ = '8>?7A@ B I+ &!- & &-21 f;/ m ']" . &-21 1 -; 0f / 3 I+ &-!& &-21 £ f / = '8>?7A@ B &-21 &-21 0f;/ 1 -; 0f;/ '#" . m £ '*>?7R@CB I+ &!- & 1 -; f;/ = m '#" . £ ¤ &-21 0f / a 1 0f / I+ &-21 a 1 f / ( 7X! ' m m . a D -21 & -21 F GIH

+ f F & 2 1 1 &-21 G '*>?7R@CB = % ' & ( X 7 ! m m . D -21 1 -; F GIH f &-21 - + a 1 &-21 F G = '*>?7R@CB m m D -21 &-21 F G 1 -; F G9H ' m ¤ ( 7X! f &-21

- + a 1 . 1 -; F G

m

28.

29.

30.

,
h7A@CB

G ¤

+ (

11.

. EDI+ &- . D + &-21

12.

13. 14. 15.

16.

17.

m F -;&-21 m F

-;&-21 3 / *F

7 m G9H F G &-21 -;&-21

m F G ¤

G9H

m a 1

m G -21 a 1 F G ;- &-21

m ! '#" ¢ a &-21 1 -; fI& 0 3 1 &-21 m m ! '#" D -21 R 1 -; F GIH 3 - R m ']" . ¢ a &-21 1 -; 0f;/ 3 1 I+ &!- & 1 -; £ f;/

m F G ¤ = '8>?7A@ B = '8>?7A@ B = '*>?7R@CB

&-21 &-21 0f;/ 1 -; 0f;']/ " . 3 ¢ a 1 m '8>?7A@ B £ I+ &-!& & -21 f;/ = &-21 0f;/ a 1 0f;/ '#" . 43 ¢ a 1 m £ ¤ I+ &-21 -;&-21 f;/ m m m '#" . ED -21 1 -; F GIH 3 - + a 1 1 -; F G = '8>?7A@ B m m D -21 &-21 F G 1 -; F G9H ']" m ¤ . 3 m - + a 1 &-21 F G m m ED -21 &-21 F G a 1 F G9H m ¤ 3 7 f &-21 - + a & . F -;&-21 G &-21 &-21 0f / 1 -; 0f ']/ " . 3 ¢ a 1 m £ '8>?7A@ B I+ &!- & &-21 f / =

18. 19.

20.

21.

22.

&-21 1 -; 0f;/ & -21 0f;']/ " . 3 ¢ a 1 m £ I+ &-!& &-21 ;f / 3 N

= '8>?7A@ B

, 3 8 , 3 8

23.

24.

25.

26.

&-21 &-21 0f;/ &-21 0 f;/ ( 7X! ' . f &-21

I+ &-!& 1 -; &-21 0f / -;&-21 0f / ( 7X! ' . f &-21

I+ &-21 a 1

£ f / ¤ £ f;/ ¤

a 1 0f;/ 1 ;- 0f;/ '%& ( 7X! f &-21

I+ &-21 . a £ f;/ ¤ 1

m m D -21 &-21 F G &-21 F G9H ]' " m ¤ m a . - + 1 1 -; F G m m ED -21 1 -; F G &-21 F G9H ']" m ¤ 3 m a . - + 1 &-21 F G m ¤ m m m '#" a . D -21 &-21 F G -;&-21 F GIH - + & a 1 F G m m ED -21 a 1 F G 1 -; F G9H ']" m ¤ . 3 m a f

+ F & 2 1 & a 1 G

27.

28. 29.

1.11.2. Derivatives with respect to the order 1. 2.

" "' ' M ]! % 3 ¢ & ¢ & -21 $&J % ')L ( $ ! & ¤ M]! 3 0& 43 ¢ M! ¤

1 1

2?7R@CB 2 m m m 7 ' $ L F G F G ¤ G 10. D &-21 F G9H F 3 J &m F]3 m G - + . -21 F m G ¤ 11. ED -21 F G9H ( 7! ' m m . &-21 -;&-21 F m G ¤ I G H G F F 12. EDI+ &- a 1

( 7X! ' m m . &-21 -;&-21 F m G ¤ I G H G 13. EDI+ &- F F -;&-21

m . 14. D I+ &-21 a 1 F G9H m ¤ 3 ¢ / *F m G a 1 -21 a F m G F G 1

-;&-21

m ! '#" 15. &-21 &-2 1 = '8>?7A@CB &-21 0fI& m ! " '#" -;&-21 £ ¢!!£ fI& ¤ 16. k &-21 - a 1 fI& 5n'65 5n'65 1 L £ J 17. k &-21 fI& fI # -21 5_ 7X! 1 N 1 5_7 m 1 ¤ ! ' ( X 7 ! m £ fI &-21 - R '8>?7A@CB I G H F = 18. ED -21 R

&-21 m ! " '#" £ ¢) m ¤ m ¢ G F 19. D -21 V - R a 1 H 43 7 m 20. ED R F GIH 5n'65 #$ 5n'65 1 ¤ 3 ¢ £ fI # - -;&-21 J 5_ 1 7XL! N 1 1 5_7 ! 7.

3 3

+ (

21. 22. 23. 24. 25.

26. 27. 28.

29. 30.

m ! ']" &-21 0fI& &-21 0fI& &-21 £ fI& ¤ m ! '#" &-21 0fI& &-21 0fI& &-21 £ fI& m ¤ m m ( 7! ' £ ED -21 &-21 F GIH fI &-21

- m m m ( 7! ' £ D -21 &-21 F G9H fI &-21 - m ' V f / F G J $ L a 0f / -; a 0f / ¤ m '#" . £ &-21 V& -21 f;/ 0 I+ &!- & &-21 f;/ m '#" . £ &-21 V1 -; f / 0 I+ &-!& &-21 f / &-21 &-21 f;/ 1 -; f;/ ']" 0 . m £ I+ &!- & 1 -; f;/ m ']" . £ &-21 0f / a 1 f / 0 I+ &-21 a 1 f / m ED &-21 F GIH F3 mG -; -21 ' $L a F m G a -; KJ ' ( 7! m m . D -21 V& -21 F G9H f &-21

- + a 1 &-21 F G ( 7! ' m m . a

+ ED -21 1 -; F G9H f F & 2 1 1 &-21 G

31

= '8>?7A@CB = '8>?7A@CB

= '8>?7A@CB = '8>?7A@CB = '*>?7R@CB ¤ F m G ¤

31.

= '*>?7R@CB

32.

33.

m m ED -21 &-21 F G 1 -; F' GIH m ( 7X! f &-21

- + a 1 . F 1 -; G m m ED -21 &-21 F G a 1 F GIH ' ( 7! f &-21

- + a & . a 1

34.

395

= '*>?7R@CB = '8>?7A@CB m

F G ¤

35. 36.

37. 38.

39. 40.

41. 42.

0fI& F m ED &- - R PED a &-21 R F GIHOH F 7 G 7 &- P - 0fI& F 3 a - P -

7 G £ fI - 0fI& ¤ m £ fI - ;- &-21 -R F G ¤ G 3 £ fI - fI& ¤

m ED a R ED &- 2- 1 V-R F GIHOH F 7 3 G 3 £ fI -;&-21 R F mG ¤ 7 £ &- - @CB $&J % ')( $ ! H- & = Q

3.

4.

5.

M]!

3 ¢ a 1 0& #' " ; - "' ¢ 4 3 &

! _!¡

H

L

H

"" ' ' % & -21 ( L 3 J$&% ' ( $ !

H M! 1 * Y C £ & 3 £ & 0

£ & 3 £

H

- & = Q> @CB

0& = = @ B #O !C@CB

H M! *

1 Y £ & 3 £ & 0&3 £ & @CB 9 $ ( ! '#" "' L M]! ¢ 3 &

¡\ !

; - ""' ' % & -21 ( L 3 $&J % ')

( $ ! L- 0& = Q> @CB 7 7 L M]! £ £ 0& 3 F

C ,G - & ?7A@ B m ! / fI H = &-21 D m m 10. ED &-21 A F

0 I f & 3 &-21

&-21 fI& G9H '#" ! O ) ' 8 ( 7 ! m £ '8>?7A@ B / fI H = &-2 1 D m m

11. D &-21 F

&m -2 1 fI& m 3 &-21 fI& G9H # ' " ' (87! m ! £ fI / £ fI &-21 D / O')(87! £ fI / £ fI H ¤ / 3 m m

12. D &-21 F

&m -2 1 fI& m &-21 fI& G9H ] ' " O')(:7X! m ! £ fI / £ fI &-21 D / O')(87! £ fI / £ fI H ¤ / m ']" . f < 3 f I +

13. k &-21 / / & ! & ( O ' ! % m m -; 1 ( L ' J

. E S a j

+

S 13. ED 0 3hf F ( m GIH 1 43 ¢ 5n% '! % 3hf9

- + SEa a 1 . j SEa F ( m 3.

;
?7A@CB 0 I f & = -; 7 ¢ F G -;&-21 0fg3h& ¤

3h

S -; m m , 3 f9

S -;

m 5 S -; ) m (8 , ! % ( O'! % f9 5 m S -; ) ( m ,

= > 'I@ B = > 'I@CB

m

D0 f 3h S S F m (8 GIH m (*! % '! % 0f 3h S -; S > I' @ B F (8 G = m

- m

a 1;D0 f9 3h

S S:F m (8 GIH ! % m (* ' 5 R7 @CB O')(:7X! % ; f9 3h

S -;&-21 S - &- F m (* G = > g ;9P

+ (

m

ED0 f9 3h

SEa 1 SEa 1 F m (8 GIH m (* 5_'g75 ! % 7! % f9 3h O S ;- a 1 S m (8 a F

- 1 S 8. ED0 3hf9 O - S:F ( m G9H ')(:7X! % . 3 5n(: 7X! % 3hf9

- + SEa SEa F ( m G S S 9. EDM &- -21 3hf9

S F ( m GIH

(*!O% '! % f9 - S -21 3 f9

S -; S ( m F

- a S S 10. 1;D &- 0 3hf9 O S F ( m GIH

! % 3 (* ' (87! % f9 a

- S -21 3hf9

S -;&-21 S - &- F

P1

7.

= > 'I@CB G

= 5n'*>?7R@CB G

= > 'I@CB

( m G ' 5_7R@CB = > 6

ED &- S - 0 3hf9

SEa 1 SEa 1 F ( m GIH ( O5_'67X5_! % 7X! % f9 - S - 0 3hf9

S -; a 1 S a F ( m G

- 1 = > 'I@ B m 5 S 12. ED &-21 PED fT3h& S:F m (8GIHOH £ - (*!O% '! % -21 ] fg3h& S -; S F m 5n * ( G = > 'I@ B m -; m 5 S a S 13. D 1 D &- -21 fg3 S F m (8 GIHOH (*! % '! % F m G - S -21 fg3h& S -; S F m 5(8G = >_'I@CB -; m m 5 S a 14. ED 1 PED -21 0fg3h& - S8F m (8GIHOH £ - £ fT3h - S -; SEa F m 5(8 G ¤

m SEa &-21 fT3h& - S S F m 5(8 GIHOH 15. ED &-21 PED m £ - £ f S -21 0fg3h& - S -; SEa F m 5(8 G ¤

m £ fI a 1 0fI& = > 'I@CB 16. ST0fI& S -; 11.

: N

17.

18.

5 m n S ; f9 3h

S ) m (* , 5n ! % (* '65n ! % ;0f9

m S S &- - f9 3h

S ) m 5nO! % (* '65nO! % f - S - 0f

T n (z )

3h

S ;- S ;- 5n (* , 3h S -; S -;

U n (z )

m 5 ) m (8 ,

= > 'I@ B

m 5 ) m (8 ,

= > 'I@CB

m

D;0f h 3 S S F m (8 9G H m (*O5_'675 ! % 7! % ; f9 3h O S -; S F

- m (* G SEa 1 SEa F m (* G9H 20. ED;0f9 3h

1 m 5:O! % m (* O'65n ! % ; f9 3h

S -; a 1 S - a 1 F m (8 G S 21. D 0 3hf - -21 S F ( m GIH 43 ¢ 5n% '! % 3hf9

- + SEa . -21 SEa F S S 22. ED &- - 0 3hf9

S:F ( m G9H

( O5_'67X5_! % 7X! % f9 - S - 3hf9

S -; S

- F ( m G S - & 0 3_f SEa 1 SEa F ( GIH 23. D &- ! m

1 ( O58' O58! % O! % f9 - S !- & 0 3_f9

S -; a 1 S a F (

- 1 19.

24.

25.

26.

m 5 D & -21 P D &- S 2- 1 fg3h& S S F m (8 G9H H m S m 5nO! % * S ( 'g5:O! % F G - -!&4 0fg3h& -; S -; F m m 5 D a 1 D 0fg3h& S S F m (8 GIHOH 5:O! % m £ - (* O'65n ! % 1 fT3h& S -; S -; F m m 5 D a 1 D SEa a 1 0fg3h& - S - S F m (8 GIHOH 5:O5_'67X5_! % 7X! % F m G SEa 1 fg3h& - S -;&

:1

= > 'I@ B = > 'I@CB ( m G ¤ = > 'I@CB m G = >:'I@ B

5 (8G

= > 'I@CB

5n (* G

= > 'I@ B

m 5 ¤

ES a F m 8 ( G

+ (

27.

,9N1

m 5 ED &-21 PED 1 0fg3h& - S - : S F m (8GIHOH ' 5 7! % ¤ £ - 5n 5_g S -;&- ES a F m 58 7 ! ( G % g f h 3 & m

1.20. The Hermite Polynomials Hn (z) 1.20.1. Derivatives with respect to the argument

% £ * ( '! % fI S -; 0fI& = > 'I@CB 1. ST fI& 0 2. -21

7

ST0f / 4 3 ¢ SEa £ S _ F 3 G -;&-21 S-;&-21 0f9

& ¤ 3. SEa 0f;/

1 3 ¢ SEa £ SEa 1 ? F3 7 3 G fI -; a 1 -; a 1 0f & ¤ S ! % S S T f;/ 0 (* '! % - S -21 S - 0f;/ = > 'I@CB 4. &- -21 S 5. &- -!&4

SEa 1 f / 0 5_7X! % (* '65_7! % - S -!&4 S - a 1 f;/ = > 'I@ B m % m ( '2! % 43 £ fI -;&-21 S -; F G = > 'I@CB 6. &-21 S F G m S S:F m GIH 43 ¢ ( !O% '! % S -; S = > 'I@CB 7. EDM

- F G SEa 1 SEa F m GIH 8. DM

1 m 3 ¢ (* 5_'67X5_! % 7X! % S -; a 1 S > 'I@ B a F

- 1 G = 7 m m ¤ ¢ £ _ S S 9. DM &-21

F 3 G -21 S-;&-21 ) ,

S F GIH 3

m 10. ED &-21

SEa 1 F G9H 3 ¢ S £ SEa 1 _ F]3 7 3 G fI -!&4 -; a 1 m ¤ S ) , W W W W 11. ED - S 0fI& H 3PfI - SEa fI& ¤ W W W W m m S F G9H f -;&-21 - SEa F G ¤ 12. D &-21 - :,

13.

14.

15. 16. 17.

18.

19.

20.

W

# . '/

ST0f;/ H 3 ¢ S £ S -I&-21 - W -;&-21 f9

& ¤ ES a W ED V- SEa 1 0f;/ H 3 ¢ S £ SEa 1 fI -I a 1 #- W -; a 1 0f & ¤ SEa ' W W ( X 7 ! D SEa &-21 - ST f / H ' S -21 - SEa 0f / ¤ W W ( 7X! ' S E S a ' V- SEa a 0f;/ ¤ D V- SEa 1 0f;/ H

1 W m D &-21 - S F GIH 3 ¢ SEa £ S -1 - W -;&-21 m ¤ SEa ) , W m ED &-21 - SEa 1 F GIH 43 ¢ SEa £ SEa 1 fI - &4 - W -; a 1 m ¤ SEa ) , W m S D - -21 - S F G9H 7 ' - S -;&-21 - W SEa F m G ¤

W m ED - S -21 - SEa 1 F G9H 7 ' - S -;&-21 #- W SEa a F m G ¤

1 ED -21 -

1.21. The Laguerre Polynomials Lλn (z) 1.21.1. Derivatives with respect to the argument 1. 2. 3. 4. 5. 6.

(Q(*')(:7 '65_7 c & c 0& c & 'I@CB

¤ Sc -; 0fI& Sc -; fI&

= > I' @CB = > 'IC@ B

+ (

m m EDM &- c -21 Sc F GIH 3Pd 3 - c -21 Sc -; F G ¤ m ¢ m S 8. DM Sc F GIH 4 3 43Pd 3 S -; Sc -; F G 43fI - c a fI& ¤ 9. - Sc 0fI& S n 5 ' ! % c -;a fI& ¤ 10. c - Sc 0fI& % c -; - SE a2SEa - Sc fI& 0 5n% '! % c a2S - SEc a fI& ¤ 11. c m m ¤ a 12. &-21 - R Sc 0 f - R Sc F G 5:'! % m ¢ G 3 % - c -21 -R SEc -;a 13. &- c -21 -R Sc F m S 14. - c - -21 #-R Sc F G 3 ¢ 5n% '! % - c - S -;&-21 -R

, ,

7.

= > 'I@CB

m F G ¤ m ¤ SEc a F G

1.21.2. Derivatives with respect to the parameter 1.

-/' M! & -21 7 ')( $

= @ B

c &

1.22. The Gegenbauer Polynomials Cnλ (z) 1.22.1. Derivatives with respect to the argument 1. 2. 3. 4. 5. 6.

7.

S c 0fI& £ fI 0d2 S c a -; 0fI& = > 'I@CB &c a2SEa &-21 c S 0f;/ 0d2 &c a2S -21 c S a f;/ ¤ c a2SEa &-21 c SEa 1 f / 0 d2 c a2S -21 c SEa a 1 f / ¤ &- S -21 c S 0f / 0d2 - S -21 c S a - 0f / = > 'I@CB &- S !- &4 c SEa 1 0f;/ 0d2 - S -!&4 c S a - a 1 f;/ = > 'I@CB 5n'! % 5nO')(:7X! % % (87! % % SEa &-21 ¢ 3hf & 4c -21 c S 0f;/ £ -; % 7 Y C7 ( &! ' S -21 ¢ 3hf9

& c -;&-21 c SE-;a f / ¤ n 5 ' ! n 5 O 6 ' _ 5 7X! % % % 5 7! % % SEa ¢ 3hf9

& c -21 c SEa 1 0f;/ £ -; % 7 Y C7 ( &! ' S ¢ 3hf9

& c -;&-21 c SE-;a a f;/ ¤ 1 :95

0 ' /

£ fI d2 -;&-21 c a F m G = > 'I@CB S -; m S c F ¢ 3 G9H 9. (* 6( ! ' m C7 ( &! ' &- c fT3 £ & c -;&-21 ES c -;a F ¢ 3 G ¤ 5 m 5 £ m n ( G

d2 f 0fg3h& -;&-21 S c a -; F m * 10. D 0fg3h& &-21 S c F m (8 GIH = > 'I@ B m 5 11. D&c -21 fg3h& &- c S c F m (8 GIH &! ( ( 43 ¢ 5n% '! % Q(*J O 1 '! ( L ' c -;&-21 fT3h& &- .c c -; F m 5(8 G ¤ &(' SEa m m 3 ¢ S d2 S -; c a F m G S = > 'I@CB 12. D c S F GIH

S - SEa 1 c SEa F m GIH 3 ¢ 0d2 S -; a 1 c S a a F m G 13. ED

1

- 1 = > 'I@ B m S 14. D - c - 0Q3hf 4c -21 c S F GIH

7 3 £ -I 5n% '! % 5nO(8')7(:! % %7X! % % C7 ( &! ' - S - c Y 0Q3hf c -;&-21 c SE-;a F m G ¤

m S 15. ED - c - -21 0Q3hf9

c -21 c SEa F G9H

1 43 £ -; 5n% '! % 5n 5_'67X5_! % %7! % % C7 (7 9! - c - S -21

' Y 0f9 3h& c -;&-21 c SE-;a a F m G ¤

1 5 m S S 16. D 0fg3h& &- -21 S c F m GIH ! ( C7 (* Q( ! ' 5 m . C( 7 9( > 'I@CB Q( ! ' (9! ( "' f

I+ S -; fg3h& - S -21 S c F -; m G = a2SEa &-21 S c F 5 m GIH S 17. ED - c - 0fg3h& c m n 5 ' ! % 43 ¢ SEa . fg3h& c a2S -21 SEc a F 5 m G ¤ %

+ f c m S 18. ED 0 3hf9

- c - S c F ( m GIH 3 ¢ 5n% '! % 3hf9

- c - + SEa . c F G ¤ SEa ( m 8.

m EDM &-21 S c F IG H 43 DM &- cI fg3 £ & 4c -21

£ -; 5n% '! % C7

:;

+ (

, , ,

1.22.2. Derivatives with respect to the parameter 1.

2.

' / M! 5n')(* $ d2 3 d2 c & $ ( 5n')( $ ! &c - & 1 = = @ O!C@CB 7 D F d 7 G 3 Fd G 3 £ £ d2 £ £ d £ H c & '#")" $ 5 9! & -21 7,5hC( 7X! £ ')( $ ! P5 $ 5:'! c & = @CB

3.

4.

£ £ d2 'I@ B 0fI& ¤

&' 3 465 78

. M ¢ 3hfI& ¢ If & j S + l5n 0fI& 43 £ If % '! % ¢ 3hfI& -; ¢ fI& -; j + -;9lV-; . fI& ¤ SEa ¢ SEa a ¢ fI& j S + l . fI& 10. M 3hfI& 43 £ fI 5n% '! % ¢ 3hfI& a2S ¢ fI& -; j + lV-; . fI& ¤ SEa . ¢ a 11. M fIQ3 j + lC-;&-21 fI& m 5_7 ¤ 5_5 77! ! ' f fIQ3 ¢ a 1

, ' ) * . k a 1 j S + lM1 - S 5 -;7 ! ¢ ( fI(:

7X ! 12. ( (*'65_ 7! ( ' - a 1 j + -;9lM1 - SEa . ¢ fI

¤ S . m 13. ED &-21 j S + l F G9H - ¢ F3 mG -;&-21 j + a 9l a . F mG = > 'I@ B S -; . m 14. &- -21 $ f9 j S + l F G . m f 4 3 83 - -21 0 fI -;2j S + a 9l V-; F G ¤ . m S S 15. D 0 fI &- -21 j S + l F GIH f 3:3 S -; 0 fI - S -21Oj + a 9l . F m G = > 'I@ B S -; S fI a a2SEa j S + l . F m G9H 16. ED - - V- -21 0 . m 43fI ¢ - - V- S -;&-21 0 a a2S j S + a 9l F G ¤ I f . m a 17. ED - - &-21 Q3hfI 0 fI j S + l F GIH . m £ fI 5n% '! % - - a &-21 0Q3hfI -; $ fI V-; j SE+ a -; 9l V-; F G ¤ . m SEa a S 18. ED - - V- -21 0Q3hfI fI j S + l F GIH £ fI 5n% '! % - - V- S -21 0Q3hfI a2S fI V-; j + l -; . F m G ¤ SEa . ¢ m 19. ED &-21 j S + l F 3 G9H F mG ¢ -I&-21 j + a 9l a . F ¢ 3 mG = > 'I@ B S -; 9.

:
'I@ B m . ED - - V- S -21 j S + l F ¢ 3 G9H 43 ¢ ¢ - - V- S -;&-21 j + l a . F ¢ 3 m G ¤ S m . ED &- -21 j S + l F ¢ 3 GIH 3 ¢ 3 ¢ - -21 j + -;9l a . F ¢ 3 m G ¤ S m . ED S 0fg3 £ & j S + l F ¢ 3 G9H m . £ 3 83 S fg3 £ & -; j S + l V-; F ¢ 3 G ¤ m . ED S 0fg3 £ & &- S -21 j S + l F ¢ 3 G9H . m f 4 3 83 S -; 0fg3 £ & - S -21 j S + -;a 9l F ¢ 3 G = > 'I@ B m . ED S 0fg3 £ & &- S - -21 j S + l F ¢ 3 GIH m . 43 £ 3 g3 S 0fg3 £ & - - S -21 j S + -;9l F ¢ 3 G ¤ m . D - - V- S -21 0fg3 £ & j S + l F ¢ 3 GIH £ 5n% '! % - - - S -21 fT3 £ & V-; j + l-; . F ¢ 3 m G ¤ SEa m . D &- - V-21 fg3 £ & j S + l F ¢ 3 GIH £ 5n% '! % &- - -21 0fg3 £ & V-; j + -;9l -; . F ¢ 3 m G ¤ SEa m . ED - - V- S -21 0fg3 £ & a2SEa j S + l F ¢ 3 G9H £ 5n% '! % - - - S -21 fT3 £ & a2S j + -;9l . F ¢ 3 m G ¤ SEa . m 5n ED a 0fg3h& j + lC- -;&-21 F m (* G9H 5 5_77X! ! ' f D a 1 F " ¢ 3 G9H ¤ ' m . 5 m D - -;&-21 63hf9 2j + lC- -;&-21 F E( m GIH 3 ¢ 5_5 77! ! ' - -;&-21 D a 1 F " ¢ 3 mQG9H ¤ ' : L

K

E

D

1.23.2. Derivatives with respect to parameters

' 3 465 78 M]! . £ ¢
7 @ @CB = ! =

(*')( (

7 @ @CB = ! =

!

1.26.2. Derivatives with respect to parameters

m ' 5! 7 L H SEa D1N 1Jg 5n% ' ' % ! %

1.

a 1 ¢ ?7R@CB 3O 3P& - 3P& " 7#7 ¤ 7 ( N ( ! L J m! D J ' L H S 36 # J ' ! L " ( &- S -21 ')( (87 ¢ $ L 3 - (:7X! % J ( ! " ( &- S -21 ')( (:7 ¢ $ 3 (:7! % J L 3 43&

2.

3.

<
?7A@ B

!

= '8>?7A@ B

9 = '*>?7R@CB

I = '*>?7R@CB

O

= '*>?7R@CB

Z

7.

m 5 0f O m 0 f O 7 Z 3

& -21 0f O 3 m

31 , Z

m 5

f O ¤

3.2. Special Functions 3.2.1. The Bessel functions Jν (x), Yν (x), Iν (x) and Kν (x)

4-4# $ Z 3 a a 1

1. 2.

3.

4.

5.

6.

= " £ a a 1 ( 9$ 7% ! F G a a ¤ & - 1 Z £ (*'65_7 G &- a a 1

F ( '65_7 Z £ * G F a ¤ '& Z ¢ a 1 5_7X! £ X a 1 ¢ ¢ 3 ¢ X a 1 1 £ 3 1 a 1 ¢! ¤

3 ) ¢ ! ¢ ! ¡ \ ) ¢ & Z 3 3 £

£ ¤ 3 1 Z 3 £

£ ¤ 3 1 7 Z " ! " ! ! F !6G 3 N F !6G = ! !C@MB

7.

Z

8.

Z

9.

Z

10.

Z

! ! F !G

']" = !C@ !C@ ' & = " '#" = !C@ !C@ ' & 3 =

7

&-21

! F V! G ! ! D " !H ¤ ! ¤ !H ' 7 ¤

2- 1

N ' O' D

N,

= ! !C@MB

11. 12. 13. 14. 15. 16. 17.

18. 19. 20.

21.

22.

23.

24.

7 -; &-21 a 1 3 ' -; ¤ Z " ! ¤ !

Z '& ! 3 ¤ !

' !& ! Z !& !C@ ']" = O' D " ! H ¤ 2 -21 0 = " !C@ '%&

Z 7 ! ! ! 3 ¤ D !H Z 7 !C@ 7 '& ! ! ¤ = " ! !& !

" '#" Z = !C@ ! '%& ' ! !7 ( ! ! ) 7 ! ! Z 3 8 3 8 3 N ! , ! ) ! ,

) 7 Z 3 8 ! 3 8 ! ! ! ,

N ! , ) ) 7 Z 3 8 ! 3 8 !

N ) 3 8 ! , = 3 8 !C@ ) 3 8 ! , Z & -21 (65_7 G - a a 1 £ F Z (65_7 £ G F " ' (

! Z 1a - L' J

(65_7 Y Z a 3 £ &- F 3 G -; &1 Z V ?7A@CB

£ If

£ If 0&3 ¤

£ £ fI

£ / £ fI ¢ ¤

f

0f

£ If 3

£ If 0 ¤

f

0f

£ If 3

£ If 0 ¤

31

+

Z

92.

93.

Z

94.

Z

f

f

] f

f

¢ 3

£ If 3

.

+

0f f

5) ;

£ If 0 ¤

£ / £ If ¤

£ fI ¤ 4

4.1.5. The logarithmic function

1

1.

Z

2.

Z

3.

Z

1

1

(

7

(

7 7 D 3 F £ G F £ G 3 F £ G F £ 9G H ¤

7 ¤

(

3

S Z " 1 V7#7 #$ ¢ 7 m &C7 ( ! !

"

@CB

¢

@ B

@CB

B

B

B

#$ 1 # 7 # 7#7 ! (

m N & # 7 ( # 7
# 7 ( # , 7 5 ! &

" V7#7 #7V7 #$ Z 1 ! m N & V7 ( #7
Z m 5 m ( F £ f 7 G / f I

7 ¢ &C7 ( f * m 3 / f I 3 f = m ! Z m 5 m ( -!&4

/ f / f g3 7

2.

1 -21

Z

!>

m 5 & L

! 5 7J ! 5n

f

7 5_7 5 Y N 5n 5n & 3 ¢ N

m &

&

! 7 #7 #7 5 ¢ #' N & 5n /

7 #7 5 & n 5 m ! & = C7 ( m ! @ B m!

4.3. The Sine Si (z) and Cosine ci (z) Integrals 4.3.1. Integrals containing Si (z) and algebraic functions 1.

m m 0fg3 £ / f F #" G 3 £ m m m / f D 1 F #" G H F#" G 3 F #" G H1 F #"

Z

-21

/ £ 1 F#" m GIH

m

G

=m

@CB

4.3.2. Integrals containing Si (z) and trigonometric functions 1.

7

Z

£

m ( !

£

m

D fI

2.

3.

m ( !

Z

£ 3

f

4.

" " 5n ! - a & a a & J L J

Z O' ! 0f

m

m m F G m 0fI H F G

m m 3 F G m m m D fI Z

12.

!>

" " C7 ( !

7( 5 7 ( ( Y F G F G &N

W #$ ;1 - a ! ( 1- # 7 (

=

1 1;- & &

7R@CB

4.3.3. Integrals containing Si (z) and the logarithmic function 1.

Z

-21

Y

2.

Z

Z

¢ N & m 5 m (

m 5 m (

&

3.

& m 9a 1 L 5_7X! J 5_7

L J W W W W #$

#$ 9a 9a 9a 1 1! ( 1 1! ( 3 N & 5 7 9a

& & 5_7 & &

= m ! ( 7R@CB m £ f # f 0f O 3 0f O

1 ¢ 0f9 3 1 f O H 0f O 3 0f O H1 f O 0 = m @ B

m 5 m (

m ] 1 0f O £ f # f ]0f9 ¢ 0 f O 3 f9 m f 2 1 f O H 0f O 3 0 f O H1 f O 0 = m

@CB

4.3.4. Integrals containing Si (z) and inverse trigonometric functions 1.

1 -21

Z

0f

m 5_7 L ] 5_7! J 9 a & L J

3 £ f

m

2.

1

Z

&N

1 9 a 1 5_7 ! (

& & 9a & 9a &

W #$

=

( 7A@ B

£ 0f9 ¢ fI 3 £ f fI 1 ¤ f9 ] 1 fI H 0fI 3 0fI H1 0fI L 3

+

5) 31 ;

4.3.5. Integrals containing products of Si (z) and ci (z) Z

-21 £ 3 £ 0

" " V 3 ¢ 0

7?7A@ B

=m

@CB

# #

Z

21.

43 ¢ Z

-21 j ) " ¢ ' (87! % a 1 1 J L' " 7( m (

5) 5) ,

3 0fg3 ,

m m m F G 1 * j ) * , 3 j a 1 ) * ,

" ¢ 3 0f63 ,

j a ! 1 ) 43 ¢ ' % j a

1 &L ' J

7,5 m ( ! Z 3 j ) 7( m ( ! ,

-21 ¢ 3 0fg " ' 3 ' % -21 ] f 3 £ a 1 j a

1 ) * & J L'

22.

23.

Z

-21 fT3 -!&4 j

24.

W

Z

-;&-!&#-R j F G

25.

=m

@CB

m , ) *

=m

m m (* ,

=m

@CB

@ B

0f63

¢ 3 m 43 ¢ / £ J 1' L % ' j

*

"' " m ' & V -R W F m G

=m

@CB

= m

@CB

4.14.2. Integrals containing Pn (z) and trigonometric functions 1.

Z

0f j

f " 65_7X! 1 L ' 3 ¢ ' % 5 7 L J a 5_7 L - _ J J

(' ]'65 1 a 1 Y N 5 7 & 1 - _

, ,9:

0f £ £ f 5_7 #$ ! a 5_7

=

( 7R@CB

& '

" " 65nO! & L ' 0 f ¢ j a 1

43 ' % 2. a J a & L f - a &L

#$ J

( ' ] '65 5_7 J a £ & & ! Y 0f £ N = (X@ B & - a & a a & f &

'#" ' )(*'65_7! Z 1 0f G J L ' 5_7 j F 3. ' _ 5 7 a % f - - L - L J J

" # $ ( a _ 5 7 £ 1;- - 1 - ?')(:7A@CB ! Y 0f £ N =

* ( ' _ 5 7 a _ 5 7 & f - - - 1

(' '65 #$ m ! Z 1 & ¤ ¢ 4. f j 7 ( m &N

F G

&- & a ! (

SZ 0f j

5.

f ( ' ]'65 1 #7 ¤ 3 ¢ ' J 1% L ' m f £ £ N ! m f & 7 ( #7, 5

SZ 0f j

6.

f 3 ¢ m f £ £ N ( ' 7]( '65_V7?7 ! m , , ) & -21 ) m ( ! " Z " m ¢ ¢ 0f 3

j ) , =m 75 m ( ! a 1

0fg3 -21 fg3

& 5_7X! 43 ¢ m 5 &L &N

J

Z

@CB

@ B

( ' ]' _ 5 7 W 1 5 & !

=m

( 7 @CB

!

0fg3 -21 a fg3

1 ( ' ]'65_7 5 #$ '65_7! m & 5 & L 43 ¢ W &

& 5nJ ! & N

& 5n !

=m !

Z

@CB

9.

=m

10.

0fg3 -21 " ¢

) & 5_7! m 5 &L &N

J

" m ( ! Z a " ¢ , 7 5 ( !

1 ) m & 5_7X! m 5 &L &N

J

Z

11.

fg3 ,

(' ' ] _ 5 7 1 5 & (

!

fg3 ,

( ' ]'65_7] 5_7 1 5 & ! (

,9 3;

=m

=m

!

!

( @CB

( 7 X@ B

( 7 @CB

# #

5) !; ,

4.15.2. Integrals containing Tn (z) and trigonometric functions 1.

Z

0f

f " 43 ¢ - 5_7 L J

Y N

2.

£ g5 7 ! f a 5_7 L 0f £ J ( ' ' a 5_7 #$ 1 ! & 1 - 5 7 a 5_7

=

( 7R@CB

0f a

f 1 " ]" '65_7! g5n ! £ 3 ¢ 0f a a & L f £ - a &L J (' J '65_ 7 5_7 a #$ & ! Y N = ( & - a & a a & &

( ' ]' #7 #$ m ! Z & ! ( ¢ f 7( m N &

F G

1 &- & a

Z m ! a F ¢ G

< 7 5

1 ( ' ]'65_7#7 & ! ( 7 ( m ! N m & 1 &- & a

'#" ' )(*'65_7! Z 0f F G a - 5 7L f - - 5_7 L J J

" #$

( £ 1;- - a 1 - 5_7 ! Y 0f £ N = ( 0f & 7 (*' - - 5_7 a - 5_7

SZ 0f

f 3 ¢ m f £ £ N 7( ( ' ]' ##7
( ' ] ' #7 ( £ £ & N 7 ( # 7,! 5 ¤

G

m

f a F ¢ 7,5

1 0f m f £ £ m f 7 ¢ f ) 7

" ¢ 7

#$

!

@CB

m

4.15.3. Integrals containing Tn (z) and special functions Z

0 f a 1

1

1.

2.

1

Z

3.

1

Z

7! m m '65_ D V;- &-21 F G 3 # a 1 F IG H m = m

3 fI 1

-21 3hfI -21 j a 1 / E F#" m G £ f ¢ h

@CB 7R@CB

m = h

-21 3hfI -21 j a / E a 1 F#" m G £ f a 1 ¢ 3hfI 1

= hm

7R@CB

4.16. The Chebyshev Polynomials Un (z) 4.16.1. Integrals containing Un (z) and algebraic functions

j ¢ 3 £ f9

0f9 32

-21 3 ¢ P

1.

Z

2.

Z

3.

4.

0f 3 -21 3 -;&-21 F G

43 ¢ 3hf9

-;&-21 j a ¢ 3 m ,

1 ) *

=m

=m

!

m

@ B

7R@CB

&L ' 1 C7 ( ! ' & 3 ¢ J6 '

5_7X! % f - &-!& f9 ¢ -;&-21

5 m

= m @CB C7 ( ! Z1 ' 3 ¢ ' % J & '6L 5n ! f - &!- & f9 ¢ -;&-!&4

a

5 m ' &

1 = m @CB Z

,!5)

# #

5.

Z

-21 0fg3 -21 fg3

5) :1 ,

43 ¢ j ¢ 3 m , )

=m

@CB

4.16.2. Integrals containing Un (z) and trigonometric functions 1.

Z

Y

0f f

f £ £ 0f

43 ¢

( ' ]'65_7 a J

1 N & 1 - 5_7 a

" 65_7! 5 7L a 5 7L J 5 7 #$

! = ( 7A@ B _ 5 7

0f a

f 1 " '65_7X! g5:O! m 0f £ f £

' % a a &L - a &L J ( ' '65n 5_7 a #$ J

& ! Y N = ( @CB

a a a & & - & &

Z 3. f F ¢ G

£ ¢ m ! N (' 'g5 7 #7 ! ( ¤ 7 ( m &

&- & a

Z m ! a F ¢ G

4. < 7 5

1 £ ¢ 7 ( m ! N (' 'g5: #a 7 ! ( ¤ m &

&- & '#" (*'65 7!

Z 0f F G

5. a - 5_7 L

f - - 5_7 L ( - a J1 - 5_7 " J#$

£ ; 1 0 f ?')(:7A@ B ! Y = £ N

( ' _ 5 7 a 5 7 & f - - -

SZ 0f 6.

f 43 ¢ m f £ £ N (' 7 ( '65_#7,7 #5 7 ! ¤ m f &

,!5, 2.

Z

+ * '

7.

8.

SZ

SZ

0f f

£ m f f

7

f f

&43 ¢ 9.

SZ

7

10.

f 0f

11.

£ 12.

SZ

13.

SZ

7,5

7

f f

m

F m

( ' ]'65n #7 #$ £ 6 ' 5_7 1 ! ¤ £ m N & & 7 ( # 7,5 #

a 1

m f f

f 0f £

¢

f m f

f 0f

£

SZ

m

0f 0f

SZ

a 1

( ' ]'65_7 #7 7 #$ £ O '65_7 1 ! ¤ £ m N & & 7( #7< 5 #

m

F G

f f

¢

f 0f

( ' ]'65n #7 7 #$ £ 6 ' 5_7 1 ! ¤ £ m N & & 7( V7< 5 #

( ' ]'65n V7 ( #$ £ 6 ' 5_7 1 ! ¤ £ m N & & #7( V 7

G

O'9!! % ' ¢ 3 # - S m , ;9P

#$

¤

# #

17.

18.

SZ

SZ

19.

20.

Z

Z

22.

Z

23.

Z

24.

Z

"

7,5

Z

21.

-

c a F ¢ G

1 ( ' # : 5 'g5 7 1 #7 ( $# !' & ¢ '6& 5_ ¤ ! S 7X! % m 3 #- N & P 5 1 # 7( # ,7 5

7

¢

-

c

-

c a

1

"

-

"

-

5) P1 ,

c )
= m @CB

43 ¢ ' (9 ! '5 & 7 N ( ' P5n '6 5_a 7 # 7 ! % m &

D c F

-

¢

5) ,9N1

¢

D c F

¢

9! '

D ' % H N

G9H

"( 7( m

c

7 D ' 9% ! ' H N m

( ' ] '65n # 1 V7 ( #$ ! ¤ 5 1 ] #7 ( #7 2.

1

Z

¢ f;/ E K / 3 E

K

D f F m G 3 £ F m G F m G hf F m GIH ¤ 1 1

4.21.7. Integrals containing K (z) and erf (z) 1.

1 -21 0 f;/ E

Z

/ ¢ 3 E

5 5 m 5 1L ( m 1 1 1 ! J 5_7X! N 5_7 5 7 & & &

¢ 3

0f;/ E K / K

#$

=

2.

W 1 -21

Z

m 5 1L

7 5 1 5 1 m ! J 5_7! & N & & 5_7 5 7

#$

=

( 7 X@ B

( 7 X@ B

4.21.8. Integrals containing K (z), S(z) and C(z) 1.

1 -21 f;/ E

Z

/ ¢ 3

5 & 7 m L J 5 N * & J L K

2.

1 -21 0f / E

Z

/ ¢ 3

5 " m J 1 L N

5 & & J L K

W

& 5 & 5 & ! ( & 5 5

#$

=

( @ B

=

( 7 @ B

1 5 1 5 1! ( 1 5 & 5 &

W #$

4.21.9. Integrals containing K (z) and γ(ν, z) 1.

1 -21 f K / m 5 5 5 J

Z

!

¢ 3

5 5 ( m N & 5_7 5 5 1 ! 5 5 1 & 1L

, < ;

= 5 !

@CB

+

2.

1 -21

Z

f K / m 5 ! 5 5 1L & J

5) , N

¢ 3

7 5 5 m N & 5_7 5 5 1 5 ! 5 1

= 5 !

@CB

4.21.10. Integrals containing K (z), Jν (z) and Iν (z) 1.

1 -21

0 f;/ E f;/ E

Z

K

Y 0 f;/ E f;/ E

K

/

1 f;/ E

K

/

1 0 f / E

K

1

2.

Z

3.

Z

4.

Z

5.

Z

1

1

1 -21

f

1

Z

7.

Z

8.

Z

0 f;/ E

f;/ E

N & 5

5

a

/ ¢ 3 m H fI ?7A@ B N 1 J 6 £ f 3 £ If £ f 0fI 3 £ ;- £ 0f 0f$ ¤ - T 1 3 N P

+

¢ '$ L 3 J 1 m !" ¢ 9. $&% !" 10. $&% O m !" ! 11. " fI ( 7 5_7X! K3 ¢ m

F £ 3 '7 G

8.

7

m

C

;

7 F¢ 3 ' G

= '8>?7A@CB

¢

m 5 7 ! ' ' % m f ¢ ?7A@CB 5_7')! (: 7X! 5_% ' 7% ! ' ']" 3P& & -21 0& = 9.

5.2.2. Sums containing products of γ(ν

k, z)

' $ L 1 & - £ 3 1 3 & J * W #$ '& ' '65 1 '65_7 5n' £ ! ¤ ! ' & 5n'! & N a 1 5n' 5:'g5 7 5n'6 5 7 1

a 1 O'65_7 £ ¢ 2. & 3 3& nJ $ L W #$ '& 8 '& '%& '65 & '65n 5n'65_7 3 ¤ £ £ ! 5n'65_7! N ! '& & a & 5n' 5n'65n 5n'65n &

5.2.3. Sums containing Γ(ν k, z) ' 1 & - 77( ( 5n ' L ¤ $ L 4 3 & T3 1. ! J J ' ¢ 1 ¢ ¢ '8>?7A@CB 2. J $ L 3 ; T3 & 3 &-21 3 -; V- & -;-2 1 & = C7 ( ! " ' % a '65_7 ¤ ' C 7 ( * ( ' ! ! 3. & " 3& T3 ' V- J 5:'g!5 7 L J $ L ')(:7X! % ¢ ' 7 ! " 1 & 3 ! ' - & -21 & $ 4 3 L = '8>?7A@CB 4. J 1.

313

+

; 31

5.3. The Bessel Function Jν (z) 5.3.1. Sums containing Jν

nk (z)

1.

( &$'% ! " £ & - a 0& * -; 1

2.

( &$'% ! " 43 £ & - 3 ¢ & * &- -21

3.

!" ' £) n 5 ' 6 _ 5 X 7 ! $ L a a " 0& 1 F G J ;

4.

!" ¢ ' £) ' 5_7! " $ 4 3 L ; 5n6 J

FO$

' ¤ G

' ¤ G

F

a 0 & ¤

a

& W " " 5 '65 1 ( #$ n ¤ ! ! 1 N 5 ] 5n

7

6 ' _ 5 1

5.

"

( L ( L J J (*'65_7X! ( ! ' J $ L " a 0 & '

'

¤ -; &

5.3.2. Sums containing products of Jν 1.

2.

3.

¢ 4 3 ; a & 1 ( 7! ' a

nk (z)

7

£ ¢

& ?7R@CB

& - - #$ W ¤ 1 W

! -Q1 & - - #$ " ¤ 1! (

= '8>?7A@CB

+

11.

12.

13.

14.

15.

16.

17.

18.

19.

7 O' 5n'65_7 ) ' ( $ $ J 3 R

; L1

' '& L a & O '65 7 ¤

1

¢ a 0&

1 7 M : ( 7X! £ ¢ a 1 0& 3h & 0 ¤ 7 ' ' & ¤ ')( $ ! % $ 5n'65_7! % a 1 & O'65_7X! % $ 5_7 ' ¢ ¢ ')( $ ! % $ 5n'65_7X! % a 1 & '! % 3h ¤ 43 ' m !" ¢ - a 0& $ h 3 O

L 1 J & J L" #$ ( ' (')( 1 1;- - & - - " ¤ m ! ' a 1 ¢ 3h -; N ' & 1 ( m (*' #7( m (* 1! ( &L ' J

m !" ¢ ')( $ ! % $ 5_7X! % F G a 1 0& 3 ( ' (*' - - #$ ! 7 * ( '

1 & m O'! % 1A- W ¤ , N & 7 ( m ( ' & ( m (*' & W )

! -Q1 C7,5 m 5n'! " ¢ ')( $ ! % $ 5n'65_7X! % C7 ( m ( '! " a 1 & 4 3 ] 5n7 '!A ! 0& ¤ m m ' a 1 3 ¢ ' $L C7 ( m ! ( " '! & ] ' %! ' & m m " &-

J ' m ! ' ( 7! ' ¤ C7 ( * ( ' ! m

'

l

( ' 8 ( 7 & J $& & % L)" & j + -I & lC- & . F G &-!& ( L ' 7 ¤ & & l J & % 3 3 N

&

20.

21.

22.

23.

) ' * '

(*' " ( O! ' . 43 ¢ $&% OJ ') 1 (* $ L 5_ X7 ! &- a 1 & O '65_7Qj + 1 lC-;&-21 ¢ 3 # O ¤

¤ £ S J $ L SEa &- & & & O'65_7 £ $ 5_7 L & - &- a 1 0& J O' ']" " (:7 $ L O]! " &- 0& J

O'65_7 ' '#" 0& ¤ ' ! ']" 1A- &-21 &

= '8>?7A@CB

5.8.2. Sums containing products of Tm+nk (z) 1.

2.

')( 1 L " ¢ J

43 ')( $ ! % $ 5:'! % J ]' 7 ! % ¢ 43 a 43 ¢

( ' L " 0 &

& * 7

( 1 L ' 1 L ' j 0& 3h j &-21 & 0

J J

7(8 7?7A@ B

7,5 , G

(')(* '65_7 ¤ 1 1 ! 7(8

5.8.3. Sums containing Tn (ϕ(k, z)) 1.

2.

3.

¢ ' ¡ $ 4 3 L T S & J ' ¢ 3 J $L SEa & S £ &-21 ;3P& a ¢ ' $ L S S:F ¢ $ G 3 ¢ S _ S 3 £ S -21 4 3 l J 1 3 31

= 'I@CB

S

! " a ' S Q ¤ $ 5n = '8>

>?7A@CB

+

4.

5.

6.

7.

8.

9.

¢ ' / 43 J $ L S ¢ ' 43 J $ L -21

1

1

1

10.

11.

12.

13.

¢ S £ S SEa _

SEa 1 / 4 3 1 S l 3 43 ¢

5.8.4. Sums containing Um+nk (z) 1.

= '8>

>?7A@CB

¢ S £ = '*> @CB

' S $ L S F $ G ? S l 3 £ S 2- 1 S = '8> >?7A@B ' SEa _ $ L

E S a F 1 1 $ G £ ¢ S l 3 £ S SEa 1 = '*> @CB ' 43 ¢ J $ L S S:F" ¢ $ G 43 ¢ S _ S l 3 £ S -21 S = '*> >?7R@CB ( & ' $ ¢ 43 J $L $ 5n SEa 1 F#" ¢ $ G 3 ¢ S ? S l 3 #& S = '*> @CB ' = '8> @CB 43 ¢ J $L & S S ) * $ 5n , _ S l 3h S 1 ' ¢ E S a

5 1 43 J $ L &

SEa 1 )+* $ , 1 43 ¢ SEa 1 _ £ ¢ A/ S 3h SEa 1 = '8> @CB l ' $ 43 ¢ J $ L & S S ) * $ 5n , 3 ¢ S ? S l 3 3P& S 1 = '*> @CB $ ' 43 ¢ J $ L -21 & SEa 1 SEa 1 ) * $ 5 , 43 ¢ S ? S l 1 3 £ ¢ X3P& S = '8> @ B

¢ 4 3 J 1 ¢ 4 3 J 1

3 ¢ S ? £ S 2- 1 S S l

; L1 5

" ' 5 7 $ L & J $

7?7A@CB J

5.8.5. Sums containing products of Un (z) 1.

2.

¢

X & 7 7(8 £ & 0 3 £ ¢ X £ 4 0& a ¢ & 5h' L " ¢ J

43 '8( $ ! % $ 5h' 5h ! % & ( ' L " 0& J ' ' % C7E(_ ! " ' 5h ! % ( 1 L' j 0& 3 J

5.8.6. Sums containing Un (ϕ(k, z)) 1.

2.

1 0 & ¤ a 1 0& j a 1 & 0 ¤

¢ ' & ¡ 'I@CB S $ L 4 3 = J S ¢ '$ L E & 43 £ & £ & a a 1 Q ¤ 4 3 S a S a J 3 3;

+

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

¢ ' 43 J $ L S ¢ ' $L S 4 3 J 1 ¢ ' 43 J $ L 1

; L1 :

¢ & 3 £ & S S l ¢ 3 ¢ S ¢ S 3 £ & S S F $ G l $ 5 3h& S S:F $ (8G 43 ¢ S ¢ i S l 3 ¢ & S

¢ ' S / $ L 4 3

J ¢ ' 43 J $L -2 1

1

43 ¢ S _ £ & 4 S S l

¢ S _ £ & 4 SEa

SEa 1 / 3 3 £ 3 ¢ S ? ¢ ' S S F $ G S l 3 £ & 4 S $ L 4 3

J 1 ¢ ' $L SEa 1 SEa F G £ ¢ S 4 3

1 $ l J 1

= '8> = '8>

= '8>

@CB

= '8>

@ B

= '*>

¢ ' $ L S S #F " ¢ $ G 3 ¢ S ? £ ¢ S 3 V& S 3

l J 1 = '*> ¢ ' $ ( & ¢ $ G £ 3 ¢ S ¢ S E $ 4 3 L S a F " n 5 $

1 l J 1 £ S E a S 3 1 = '8> ¢ ' 5 , 43 ¢ SEa 1 ? S l $ $ 4 3 L & S S )+* J 1 3 £ ¢ S = '8> ¢ ' SEa n 5 , $ $ 4 3 L & 1 SEa 1 ) * J 1 £ 43 ¢ S ¢ / S 3 £ 43 ¢ S ¢ 4 SEa 1 = '*> l 3 3 :

@CB

= '8> 3 £ & SEa 1

@CB

= '8> 1 S l ¢ 4

@CB

@CB

@CB

@CB

@CB

@CB

@CB

14.

15.

16.

, '

$ ¢ ' 43 ¢ S ? £ S $ L 4 3 & S * $ 5 , ) J 1 3 $ ¢ ' & SEa 1 SEa 5 , $ $ L

4 3 2 1

1) * J 1 £ 3 ¢ S ¢ S 3 £ ¢ l " ( 7! O' ¢ f ¢ # 5 7 $ L $ f &- -21 ) J

¢ S l S 43& = '*>

@CB

X3& S = '8> @CB $ m 5_7 $ m 5_7X! 5 , ( ! ' O'65_7X!A ' m 5_7X! ¤

5.9. The Hermite Polynomials Hn (z) 5.9.1. Sums containing Hm 1.

2.

& -21

3.

4.

5.

6.

7.

"

nk (z)

2(:7X! ' ¢ ')( $ ! % $ ! % & O'! % 3 * 2(:7 , ¤ ) ')( $ ! ' '" £ ')( $ ! % $ ! % 0& & ')(* $ ! % £ & - ¤ ' " " ' % $J L m ! " & 43 ¢ m ! ' 1 - -; O ¤ ( ( #$ 1 (*' L " 1 ' L ; 1 ¤ ')J ( $ ! % $ ! % - & J ' % - N &

1 1 !&

m !" ')( $ ! % $ ! % 3P

- & (' #$ 1- - &- - ¤ m' ! % ' - N & & 7 ( m ( ' 1 ( m (*' 1 !

( ' ( ' ' ' -21 7 ' % ')( $ ! % $ ! % 0& O'! % N 1 V7 ¤ J$ L J $ L

! (' m ( m !' m !" ')( $ ! % $ ! % ! " 0& ' % ! ' N m ( (*'6! 5_7 1 ¤

3 3
?7A@CB

7 7 $ ! % 'T( $ 5_7X! % 0 f63 0& O'65:O! % fg3 3 ¢ (' ( '65 1 V7#7 ! ' Y £ & a 3 "

a 0& O'! % m (*')(:7X! N m (*' ] ! ( 7 7 $ 5_7X! % ')( $ 5_7! % 0fg3 a 1 & O'65 ! % fg3 3 ¢ ( ' ( ' ( 1 V7#7 O]! '%& Y £ & a & 3 "

6 ' _ 5 7 ! * ( ) ' : ( X 7 ! a % 0 & N m

&

m ( ' ] ! ( C7 ( m ! " ¢ C7 ( m ! " ¢ a ! O ' ! $ % 0fg3 & 43 1 % ')( m 5_7X! £ & 43 ( ' ( 'g5 m (*')(87 ( " 1 ( ' Y fT3 3 ¢ fg3 3 ¢ N ! m

& 1 (' V('65 1 m (*')(:7 m (*')(:7 N

m (* ' m (*' ! ( " ¢ C7 ( m ! " $ 5_7X! % 0fg3 a 1 & 4 3 43 ¢ a 1 '65_C7X7 ! % ( ')m (! ' 5 7! £ & 4 a 1 m 395N

¤

¤

¤

, '

(' V(')( 1 m (*')(:7 m (*' ! ( " (' V(')( 1 m (*')(:7 m (*')(:7 N

¤ m (* ' m (*' ! ( " ( 7! ' ¢ ' ")" £ 5_7! " a / & & 5_7X! ' a / £ & ¤ 7. 43 J $L ' £ & A& 3 £ ¤ / G $ L F 8. - / & J " ( (:7 ' ( $ ! % $ ! % j &- Q & 9. W W # ( ' ( ' ( $ ! £ J 1 L ' 3 ¢ N ¤ W

' % ! -Q1 (1 *' 1

" ( (:7 ')( $ ! % $ 5_7X! % j &- Q a 1 & 10. W W # ( ' ( ' ( $ ! £ a 1 J ' 1 L ! ' 3 ¢ N ¤ W

% -Q1 (1 *' &

' 7 ( X 7 ! ')( $ ! % ( 5 7! c 3 ¢ ¤ 11. " &- 0& ( P5_7! ' - c -; 0& Y

0fg3 3 ¢ fg3 3 ¢ & N 1

5.9.3. Sums containing products of Hm

"

1.

2.

3.

4.

nk (z)

' J( 1L $ ! % & 0 3# -21 -; ¤ $ L J " ( 7X! ' '%& ' J( 1L

5 7! O'65 7 1 -;

¤ J $L $ % a 1 & ' $ 43 A J L &- A& -21 & 1 £ + &-21 . 43A W £ E3 W / -;&-21 ' ¤ J $ L Q &- & 43V

395)

& A& ¤

+

5.

6.

' J $ ' J $

7.

£

8.

£

9.

£

L & L &-

a Q

1 Q

; P1 5

¢ £ a ¤

a 1 0& 43 Q

a 1 & 43 ¢ £ a 1 1 O ¤ ' 1 (*' L " J$ ! % ')( $ ! % Q &- 0& ' %

) 1 (*' L" J 7X! % ')( $ ! % a Q $ 5_

1

&- & ' '& ' ( % 7XO! '6 5_7X!

a 1 a F

1 ( 1 (*' L " J $ 5_7X ! % ')( $ ! % a 1 Q &- a 1 & ' '%& ( 7X'6! 5_ 7X! %

a 1 a F

1 5

(

, ¤

5 G ¤

5 G ¤

5.9.4. Sums containing Hn (ϕ(k, z)) 1.

2.

3.

4.

¢ ' 43 J $ L ¢ ' 43 J $ L

= 'I@CB

ST & ¡ S /

" S 43 ¢ S £ i £ & 4 -; (*') ( ( $ ! % ! $ 5n '! % ¤ a

( 7! " ' / 4 3 ¢ SEa 1 £ SEa 1 F $ L E S a $

1 J 1 S ¢ £ ¢ i £ & a 1 -; 43 S a ¢ ' $ L SEa / 4 3

J S 43# SEa a 395,

G S

( ! " ( ' ( $ ! % $ 5n '65_7! % ¤

(]! " $ 5:'! %

a ¤ S - & -21 Q

5.

, '

( X7 ! " $ 5 3 ¢

' J $ L ES a a SEa £ SEa a

1 / S ! " a a 1 $ ( n ¤ 5 ' ! % S 1 Q a -

£ S ¢ ' S $ 4 3 L S F $ G 3 & 4

J 1 S -; ¢ "'' &('

£ i £ S $ 5n'! % (*O')(* $ ! % £ & -

& 4 - 3 £ SEa ¢ ' SEa $ L E S a 4 3

F 7. 1

1 $ G 3Q & 1 J 1 S -; ¢ "'' &('

£ ¢ i £ S a $ 5n'! % ( O')(* $ 5_7X! % £ & -

& 4 - 1 3 ¢ '$ L 8. 3 f O SEa SEa F $ m 5 G J " ( ' % 5n '! % ( L" S 1 m f S a '65 J $ ! % ( $ ! % F G S - F G 1 ( J L ¢ ' $L f O SEa a 1 SEa a F 4 3 9. $ m 5 G

1 J " ( ' % 5:O'65_7X! % f SEa 1

& ( J L ( ( S L " 1 Y a F G a '6J 5 $ ! % ( $ ! % F m G S

- 1 6.

5.9.5. Sums containing Hm 1.

2.

nk (ϕ(k,

$ m ! " $ m ( ! ']")"" F $ G ')( $ ! % $ ! %

3953

¤

¤

¤

z))

' ]' " ' ( X7 ! ' % O'! % ' m ( ! F " m " $ "'& $ m ( ! ']")"" ')( $ ! % $ 5_7X! % a F

1 $ G ' '#" " '%& 1 # ( 7! ' % O'6m 5_7X! % ' ( ! a OF " m

1

1

¤

m ¤ G m ¤ G

+

3.

4.

5.

6.

7.

8.

$ ']" O')(* $ ! % &- F $ G $ & % 1 $ '#")" $ 5_7! "" O')( $ ! % &- F $ & $ % 1 $ '#")"'& $ 5_7X! "" O')(* $ 5_7X! % &- a $&% 1 $ '#" O ) ' * ( _ 5 X 7 ! $ % &- a &$ % 1 $ m 5_7X! '#" ')( $ ! % $ ! % F $ m 5 $ m 5 7! ']" ')( $ ! % $ 5_7X! % a 1 F

; P1 :

]! '#" O O') (* ! %

O]! ' ( 7X! ' G 3 O'! % O'! % A& ¤ F1 $ G ( 7! ' ! ' & 3 O '6 5_7X! % 3 '65_7X! % a 1 A& ¤ ! ']" ' >?7A@CB ')(:7X! % =8 1F $ G O]! ' 7 G '! % ' m 5 7! ¤ O]! '%& O'65 7! % ' m 5_7! ¤ $ m 5_7 G

5.9.6. Sums containing products of Hm 1.

2.

3.

= '8>?7A@CB

¢ ' ¡ 43 J $ L& ST & ( 7! " $ 5_7X! '#" 5_7X! % O'T(* $ ! % &- F $ 5_7 $ '%& O'6 5n ! % a

( 7! " $ 5_7X! ' $ 5_7X! % O'T(* $ 5_7X! % &- a 1 F '%& '65 ! % D a 43 ¢

nk (ϕ(k,

z))

G

=

a F $ _ 5 7

1 a 3

1 $ 5_7 G a 1 F a

1

G a 1) $ 5_7 G

'I@CB

(8 5 , ¤

a F 5 GIH ¤

5.10. The Laguerre Polynomials Lλn (z) λ nk 5.10.1. Sums containing Lm

1.

(z)

( ! ' ('! " '! " 5 ! 5n' ! % S ;- & $&% $ % S 0& 395 5

= > 'I@CB

# . '/

2.

¢ ' a 0& 43 ¢ $ L 4 3 Sc J $ ' 5 _ 5 7! " 3P& Sc a $ L J % 5 7 5_7!

3.

4.

5.

6.

7.

8.

Sc a -; 0&

= > 'I@CB

0&

a ¤ Sc a & - 0& ' 3 3 E

1 ' ! " m ¢ (E5 : 5 7! " Sc a & $ L 3 J ( m 58' 5:7 ¤ (E5n7! (% (E( 5n( 7Xm ! 5n7! ' N ( 5*' 5n 7 ( m 5n! 7 L '

J (]! " 5n'! % ' ¤ ( Q(*'65_7X! " Sc a & 43 ¢ % ( &! ' SE $ c L ; & a J 5n'! % ( ! " ' ¤ ( P5 5_7! " Sc a & % ( P5 5_7! ' SE $ L c 0 & a J ( 5 7! ( ( ( ' m 5n' ¤ ' m 5 5n'! " m ! " ( P5 5 7! " 43& Sc a & $ % L N

J 5_7 m ! L J ( g5 5h' 5 7X! " ' ( g5 7X! " ( 65 5 7! " 3 Sc a 0& $ L J ' % % ( g 5 5h7X! '! ( % g( g5 5 7X! 7X( ! ( c 3& c & ¤ ' & ' SEa

Sc - & ¢ ' 10. 43 J $L ¢ ' 11. 43 J $L ¢ ' 12. 43 J $L 9.

SEc a 1 0 & 3 Sc - &

¤ SEc -;a &1 -21 0&

Sc -; 0&

= > 'I@CB

' ( Q( ! " C7 (Q(*'! " Sc - 0 & ( 9! ' Sc a ;- 0 & = > 'I@CB ( 6( ! " C7 ( m ( '! " Sc - & ( 9! ' ( # 6( m 5_7 ¤ ( P5_7! ( % m ! % ( '6! 5_7 L m '

N J 5 7 # 6( m * 395;

.

13.

14.

15.

16.

' (' ( J $ L

!" m 5 m !"

58'! "

3.

4.

5.

6.

( ' ] ( ( m 58' ¤ ` m ! ( "

nk (z)

I5_7X! ' 4 ')( $ ! % (P5_7! " c 0& ( 5 7! ' c F I5_7 G ¤ ')( $ ! ' ' ¤ & -21 ¢ ')( $ ! % ( P5_7! " c 0& ' % (P5_7! ' 3 43 3 d)3 ')( $ ! % ( ' (' ¤ ¢ 5_7#! 7 L 43 = ')( $ ! % @ ( P5_7! " c & N J m !" ')( $ ! % ( P5_7X! " - c & ( ' _ 5 7 ¤ a ! ' m c - - 1 c - - ' % -; N ! & & 5_7V( m (*'65_7#6 ( m (*'65_7 ( m !' m !" (' m ¢ ) ' ( ! ! ( 5 7 ! ' % ! ' N m ( (*'65_! 7# 5_7 L ¤ $ % " 4 3 c 0& " J '( 6( '! " ')( $ ! % ( P5_7! " 3P& - c & #$ ( 7X! ' 1;- ! ( ( P5_ ' % -; N & a 5_7 # P5_7 ¤ c 1 c

395: "

2.

¢ ' ('6( ! " m ! " Sc - & $ 4 3 L J ( 6( m ( ¤ (P5 m 5 % ! ! ' ( ]! N ( 7 (%V( 6Q( ( ( #7 ( ( m ' m ' ! ( " ` & 1 ^ 5n'! % ' - Sc - 0& % 3P& -; SEc - a 0& ¤ $ L 3 2 d J " (]! ' ( 5 5_7X! " $ 5 7! Sc a 0& $ L J " 7X! % 'g 5 7 D d c -21 0& 3 % ( P5n5 '65_5_ 7X! ' SEc -2a 1 a 0 & H ¤ S 1

5.10.2. Sums containing Lλ m 1.

Sc - & ( (]! % N ( & 1 ^

-

; N1 ,

# . '/

(']")"'&

( ' ( ' (% &BB B ¤ (]! ' (P5_7X! " c & ' % ( P5_7X! ' ES a 1 N S 2- 1 7 &B B B V7 ( " 7. ` ^ ! ( 6( '! " ) ' ( $ ! % ( P5_7X! " 3P& - c & 8. #$ ( 7X! ' 1;- ! ( (P5_ ' % -; N & a 5_7 #P5_7 ¤ c 1 c

' $ ! % ' O ] ! % ¢ ' ( P5_7! " c 0& ( P5_7X! ' c a F ,G ¤ 9.

43 J $ L ¢ ' 3 ¢ c -; & ¤ $ L 10. 4 3 c a2S & SEa J ' $ 5 !% % ¢ ' ( P5 5_7! " c a2S 0& ( 5 5_7X! ' Sc a 0& ¤ 11. 43 J $L m 5 !" ¢ ' ( P5 5_7! " c a2S 0& $ L 12. 4 3 J 7X! ( ( (*' m ¤ C7 ( m ! ' 5n( P'5_ ! % N J 5 7 m (*! ' L S ( Q(*'! " 5 ! 13. a2S $ % &c - 0& ( & ' "( S ( P5n'65_7X! ( 43 ¢ C$&7 % ( $ !( $ ! % c ¤ 7 ( G $ F E S a " ( ! ')( $ ! % $ ! % 3Pd 3 Sc - & 14. #$ ( ' ( ( 6( ]'65 1 (]% ' ! % N " = > 'I@CB

( & 1 1 "

! ( ! ) ' ( ! % $ 5_7X! % 3 d 3 Sc - 0& $ 15. ( ' #$ ( ( 6( ]'65 & (]% ' ! % N " = > 'I@CB

& 1 & ! (

¢ ' - S -21 0& S- S -;&-21 0& = > 'I@CB 16. 43 J $ L S

395
?7A@ B ' S -;&-21 S -21 ( '65 #$ ( ]! " ( P5_7X! ( & a > 'I@CB ')( $ ! % $ 5_7X! % Sc & ' % N P5_7 % = & !

395P

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

' J $ L

.

)" " ( Q(* ! " a (1 Q (* (*' c S - &

( L " ( V( 6( V( 6( ! % N " ( Q ( (* ' ( & 1 ^ ! " 7 ')( $ ! % c - 0& c -; F Q3 G ¤ (' V( m ! ' ( &! ' ' % -; N 7( m (*' 0fI 4 3& - c - 0& J

J

4 3Pd 3 S&- a ')( $

; N1 3

(*' `

= > 'I@ B

m ( ' ! Q(*'65_7 L ¤

¢ 3P& - c - & 43Pd2 43& -; c -; a 1 & ¤ ( &! ' (' &B BB 1 3P& - c - 0& ' % -; S N S J Q(*'65_7 #7&B B ! B #7 L ¤ 7 ' ¤ ! % C7 ( Q(*'! " c - 0& ' % ( 9! '

( 1;- V( ( 7X! " ( ! ' ')( $ ! % (Q(*'65_7X! " c - 0& ' % ( &! ' & N ¤

" ' ¤ ('! " & -21 ')')( ( $ ! ! ( ' &! ' ; $ % % P 3 & (Q(*'65_7X! " 43& c - & ( m !' m !" (' m V( ( " ¤ ')( $ ! % ! " - c - & ' % ! ' & N 1 m ( (*'6! 5_7 ` ^ " ) " ')( $ ! % m ( $ ! c - 0& (' #7 ' % ( &(*! ' '! 3P& -; N * ( 6 ' 5_7 #Q! (*'65_7 L ¤ m m

J )" " ' % ')( $ ! % $ ( m ! & - & ( m ! ' & 3P& -; -; 0& ¤ m 5 1 L' '( 9! ' ( ! "' 1 ( m (*' L" J ' % ( J

m ')( $ ! % & (*' L " - c - & 1L' J

J

(' m (' m Y £ fg3 ¢ N ! ! 'g5 7 ¤ N # 6 * ( 6 ' _ 5 7 5 # 6 ( ( m m

1 1

3;9N

# . '/

( Q ( 1 L" J

31. c - & * ( ' ) ' ( ! $ % & " L J ( &! ' (]! "' £ ¢ ¤ a #( Q(*'65_7X! ( 1 L ' d c -;&-21 & c -; 1 & J

' $ L 0& SEc a 0& ¤ 32. c a2 S J ' - c a2- S & $ L I f 33. J ( ( ¤ ( m 5_5n7! '' ! % (]! N ( (*( ' m ( ( ' V((' 6 " ` m & 1 ^ ! ( P5_7X! ( ' ( 9! ' " ' ' (6(*'65_7X! ( SEc - a 0& ¤ $ L 3 2 d & 34. c a2 S J ' ( Q( ! " $ # & L 35. Sc - & ( * ( ' J 1 L " ( J

( ( (*' V'( 6( ( ! ( (*O' ( " % N ` = > '8>?7A@CB & 1 ^ ! ' 5 & A ! ( 9 ! ( &! ( & $ " a ( 5 _ 5 X 7 ! ( P n 5 6 ' _ 5 7 ! ( P n 5 '65_7! ( Sc a &

36. " " Sc - 0& J $L = > 'I@ B " ( 5 7! ( ( 5 5_7X! " Sc a 0& 37. % ( #$ ' ( #$ c 5_7'g5 c 5_7 ( ! ' & ( ! '%& ! ! Y N ( P5_7X! ' & N '65 5: '65 5n

c 5_7 3 c

= > 'I@B W " ( ' J ' L 5 5_7! Sc a & $ L 38. ( * ( " J 1 L" J

( P _ 5 7 ! ( V( (*' ( > 'I@CB ! % N

J ( (* ' # 5_7 L = " ( ' m ( J L ( 5 7! ' a ! 7 ¤ ! ' % N m (:7# 1 5_ 39. $9% m " &c -

& 3;

.

40.

41.

42.

43.

44.

45.

46.

47.

48.

; N1 3

(' c * ( ' #$

7 # 6! ( O' ¤ ( g ' 5 c

7 F]3 G c - & F G F G

c - R a 1 & ¤ ('9! " ( O'! " F]3 G c - & ' m ¤ m ' ! % ' '( ! 9! ' - N (# 6 ( O'6! 5_7 L m ' m

J ( 9! " 'g( $ ! % m ! " - c - & #$ (' a c - a c - a 1 ! ¤ ' ( % 9! ! ' 3P& -; N m ' & & c a 1 ( ' c ( ' 5n7 m 5 ( '

' ( &! " 'Q( $ ! % c - & ( 5 4 , 5 5 ¤ 3 c - , c - ) , ) '( 9! " )" " ( &! '%& £) ')( $ ! % ')(P5_7X! " c - & ' % 43& -; ¤ 3 2 d J m 5n')( 1 L)" ( &! " G c - & ')( $ ! % m ! " ] F 3 ' 7 % N ( O' O'65n( m " (:7V( ` ¤ m! & 1 ^ O')( $ 5 m !A '( 9! " ')( $ ! % C7 ( m ( '! " - c - 0& " ' (' c - (*' c - a 1 (*' #$ '( 9' ! % ' m !

( m (*O! '65_7 ¤ m ' & N & c a 1 (*' c (*'65_7# 6

C7 ( m (*'! " ( &! " £) % ( 5n'! " 43& - c - & 3 d2 m ( 9! ' (' m (:7 & m % m ! ' '( % & ! ' 3P& -; N J m # Q(*O' ! L ¤ m ( &! ' 3P& - 4 P 3 2 d ( Q( 1 L" J ') ')( $ ! % J1 ( m ( ' L" ') ( $ ! % C7 ( m

('9! ' c - 0& ' % ;- N

3; ,

49.

50.

51.

52.

53.

54.

55.

56.

# . '/

$ ( ( 9! " ')( $ ! % ')(P5_7X! " - c (6(* $ ! : ( 7 &! '%& ' (Q ( * ( O'! (:7 3P& -; %

& ( ' #7 N c a 1 ( ' c ! a & (*' ¤

1 ( m L " ('9! " J ')( $ ! % C7 ( m (*'! " F]3 G c - & ( ('9! J 1 ' m % L ' ! ' F]3 G N ((*O ' ' m Q(*(*' ! '6 5_7 L ¤ m '

J m ( &! " £ - c - & ) ' ( ! ! ( % ( 6 * ( ' $ % m " & m L" J

(' m 5 5:' (87 ( m (%6(*' C ( & ! ' ¤ 6( 'g5 7 ( 1 ' % m ! ' m 5 6( 1 L ' F G & N 1 ! J

C ( & ! " £ - c - & ) ' ( ! C 7 ( * ( ' ! ( P 5 $ % m m " 1 L " J

( & ! (' ] m ( P5n' #6( m ( '65_7 ¤ ' G Q (*'65_7 ( " F ` ' % m ! ' m (P5 1 L ' &N 1 ^ ! J

m ! " ! " ( &! " £! ')( $ ! % C7 ( m % ( 9! " C7 ( ( &! " ') ( P5_7X! " - c - & 3 2 d ! ' '( 9! ' & ( ' V7 ( m ( % ( ¤ ' % C7 ( m ! ' ( & ! C 7 ( ( & ! 7 ( ( ' V 7 ( (*! ' L P 3 & N ; ' ' m m

J W " ( ' J ' L 5 5_7! Sc a & $ L ( * ( " J 1 L" J

( 5 7 ! ( V( ( ' ( > '8>?7A@CB ! %

N J ( (*O' # P5_7 L = ' 7 $ L F G 3 d 3 F]3 G c a2- S & J ( (]! % N ( ( O' 5 5n7 '6( 5_" 7'( 6( ` ¤ & 1 ^ ! 7 7 $ @& !C@CB ')( $ ! % (P5_7X! " c - 0& (P5_7! ' c &

$ L = = J

3;93

.

57.

C7 ( m (*'! " ')( $ ! % (P5_7! " - c - &

; N1 3

V7 ( m (* ' ¤ m ! ' O '( ! % ! ' ;- N ( Q' (* '65_7 ( L

Jm !

58.

59.

5n'65 1 L" ' JP ( 5 7! " £ c - 0& $ L J

P5 1 J 5_ 7XL ! ' ' C7 (* m (*'! " 1 L " ')( $ ! % C7 ( m (*'! J " 5_7X! "

60.

61.

62.

63.

64.

( ' ('6( O' V(6(* ' ¤ ] ! ' O'! % & N 1 (6( ' ( " ` ^ ! c & W #$ ( ' 1 ( m ( ' ! ( & ! ' ' m ' % m ! ' 43V& -; N & m 5 a ! 5_7 ¤ 1 c - 1 c -

Q( $ 1 L " ( c L " '65 1;- c L " ')( $ ! % J (*J ' ')J ( 5_ 7 F G c - & c L" 1 J L" J

' % ( £ 3 ¢ a 1 J c L '%& F G D + c -21 . - F ,G9H ¤ 1 J L' O' ' ( $ 'I@CB ( ' S 3;!5

# . '/

65.

m !" 2d 5 7 L " ( &$ % c ( m _ J

66.

c a J 5n L '6" 5_7X! (9! " "'& &c - & ( ' a ( #$ c 1 m! ( 5 ' % 7! ' N

c a 1 Q (* m 5_7

) ' * ( ! ( ' ( $ % 1 " L 43 ¢ £ 3 ¢ J%

- &-21 & & $ &-

£ a 1 F 7 G -;&-21 F G a 1 (' c & #$ ( &! '%& cL" & 0d ) $&% ( 5n ¤ ! a J & '65_7X! " & c & ' % N

c c & a 1 &!- &

C 7 ( * ( ' ! n 5 O ) ' ( " m m 1 " L ')( $ ! J % ( 5 7! " F G c - & & ( ' ( #7( m ! ' m 5:O')( 1 L ' m ( ' V'( 6(* ' #$ m 1 £ 'J ! % ( P5_7X! ' & & N

m (:7 ] (* m ( O' ( ! ! ' ( 5 " m m 1L" ')( J $ ! % ( P5_7 ! " F]3 G c - & & m ! ' ')( m 5 1 L ' ( &! ' J '! % F3 G ( ' m ( O' ( m (*' Y N ( ' V7 (* (*1 ' Q(*! '65_7 m

& & m ¤

¤

67.

68.

69.

¤

¤

nk 5.10.4. Sums containing Lλ m pk (z) and special functions

1.

2.

¢ ' ¢ 43 J $ L c 0& 3 ¢ ¢ c -; & & -21 ')( 7 $ c -; & ¤ C7 ( m ! ' C7 ( m ! " ( P5_7X! " fg3 c 0& ' % ( P5_7X! ' ')( m 5_7! 3P& (' m (*')(:7 V('6(*' Y 3hf- ¢ fg3 3 ¢ N ` m (*' ! ( " & 1 ^ (' m (*')(:7 m ( ')(:7 V('6(*' ¤ 3 N ^ ` m ( ' m (*' ! ( " 3; ;

.

; N1 5

( 7! " ')( $ 5_7 ! % (P5 7 ! " 0fg3 c & 3. '& ( 5 7 7 ! fg3 3 ¢ '6 5_7! % 3 ¢ c & '& a 1 ' (' ('6(*' #7V7 ( ! 1 ¤ ' % ( 5_7X! ' m (*')(:7X! N

m (*' )" " ('9! ' ) ' ( ! ' % 3P& -; fg3 $ % 0f63 c - & 4. ( &! '#" 7 (*' V7#7 3 3 ¢ ')(:7X! % m ( '! 1A-; & N & J m (*'65_7 #Q(*'6! 5n] L = >_'I@CB ( &! ' )" " ')( $ ! % 0f63 £! c - 0& ' % 43& -; 5. 7( ' V7#7 '2 Y 0fg3 £ m (* '!A(Q(*'65_7! N ! 5n * ( 6 ' 5_7Q(*'6 & &

7 (*' V7#7 '2 * ( O 6 ' _ 5 X 7 A ! ( Q * ( 6 ' _ 5 X 7 ! m N& & a & (*' # 6(*'6! 5n ¤

" ' m ( O' ( ( 7! " ')( $ ! % 0fg3 1 & & - 3P& 3 ¢ m (*' 1 N 1 m ( '6! 5_7 L ¤ 6. J ( ! ' ¢ ( 7! " a ' % 3 d2 d 3 _ & ¤ & $ % d & P 3 & 7. c -; & C7% ( 9! ' ( 7! " ')( $ ! % d 3 1 & c - 3P& O' % 0d 3 _ & ¤ 8. " / £ c 0& 9. ')( $ ! % Q(*O'65 & L " c - a a 1

J

( 7X! ' £ ¤ ' % ( Q( 1 L ' c a a 1 / J

" " 5 8 ( 7 j Q c a 0& 9 $ % 10. & " ' (' ( ' '( 6(*' #$ ' ! '! % (87 N ¤ W % : ( 7 ( * ( ' , & 1 ) 1 ! - W -Q1

-Q1 3;9:

11.

12.

13.

14.

15.

16.

17.

# . '/

$ 5_7 7 ( $ 5n6 ' 5_7X! % j Q a 1 & ' % F G ¤ & ( ' 1 $ 5_7 7 a

5n'65_7X! % j Q 0 1 & ' % N 7#7 ( ¤ $ &! ' $ 5_7 ( 7X! $ 5n'65_7X! % a 1 Q a 1 0& '! % F 63 G ¤ & $ 5_7 ( 7X! ' a $ 5n'65_7X! % 43& Q 1 0& O'! % / ¤ & 5 ! ' a ¢ '( 6(*'! " 5 G ¤

c 1 $ ! % 4 3 Q F c & P 5

& 1L ' J

¢ '( 6(*'! " $ 5 7! % a 1 Q &c - & 43 ' & 5 ]! c a a 1 F 5 G ¤ 5 1

1 L '%& J

( P5_7! ' (' 5n O ¤ . ( ! " $ 5_7X! % j + - lM1 c a 0& ' % N

# 5_7 ! L J &

nk 5.10.5. Sums containing products of Lλ m pk (z)

1.

2.

( 7! " ( 7! ' O' @ @CB ')( $ ! % (P5_7! " c 0& ( 5_7X! ' ' L c -; 0& ==

J ( 7! ' " ' ¢ $&% " " O')( $ ! % &- -21 0& ' (*' -;&-21 #& ¤ 4 3 1 J L ¢ $&% ')( $ ! % - c - & - 0& 4 3 (' a (*' a (*'65_7 ' ( 9 ! ( ! ' ' c c 43 ¢ - ¤ ! ' % & N & Q(*'65_7 )(*'65_7 P5)(* '65_7 $&% ')( $ ! % - c - 0& c - 43& #$ ' ( 1;- ( ( 9 ! ' % - N ¤ !

& c - a 1 c - 5 7 # 6( 'g5 7

3.

4.

3;
'I@B

J 5n' ( ]! " a 0& c a2SE a & 9 $ % = > 'I@CB 16. Sc J L SEc a & &3;9L

# . '/

17.

J

18.

19.

20.

21.

22.

23.

(]! ' a > 'I@CB ' % Sc & $ L SEc - a 0& -;&- c 4 3& = -; & ( ]! " a 5n' > 'I@CB $9% Sc 0& c -; a & J L SEc -;a 0& = & ( 6( ! ' 5 $ S > 'I@CB ' % Sc -; & = J $ L SEc - a 0& &- - - c -21 43& $9% (Q(* $ ! '( 9! " ')( $ ! % ') ( 5 7! " 43 Q& - c - Q c - & 3 43Pd2 a Q& -; c - & ¤ 1 ( & ! " £) d2 ( P5n'65_7X! " F]3 G &c a - Q - c - & ( 9!' ' % & F ¢ G ¤ m 5n'! " ( &! " £ ) 3 d2 ')( $ ! % $ $&! %% ! C7 ( P5n'! " C7 ( m ( Q(*'! " Y - c - 0& c - 43& (' '65 V( m c 1; - c 1 ( #$ ¤ O' ! m % ! ' ! ( &! 5 '%& 9! N '

m ' m m 5 1 V( ! "

$&% ( 9! " £ ) d2 ( P5n '65_7X! " F 3 G - c - 3 Q - c - Q c a & & ( ' c c a 1 ( 9!' ' % & N " ¤ & 1 !

5 $

nk 5.10.6. Sums containing Lλ m pk (ϕ(k, z))

1.

2.

3.

¢ ' 43 J $ L Sc ¢ ' 43 J $ L c & ¢ ' 43 J $ L Sc &

= 'I@CB

& ¡ ¤

S ( ! " ¤ ' % % (( P5 5_77! ! ( -; 5 ' L ( 5:g : ' 5 7 ! " ' a J $ 3;9P

.

¢ 4. 4 3 1 ¢ 43 S

' J $ L S Sc F ' % ( P5_7X! ( % ( P5_7X! ( "'

( ¢ E S a $ G 3 1 % S -; 5 ' S -; a J $ : S & a 7 G c - & ' m 5

; N1 :

L 3 3 d2 -

= > 'I@CB

(]! " ' ¢ a a Q ¤ n 5 ' ! $ $ % L 4 3 5. c SE Sc a J m $ m 5 ! ']")"" m ')( $ ! % F G c -; F3 m G ¤ 6. F $ "" a ( 5:O! ']" ')(:7X! % = '8>?7A@ B 7. $&% &c - & 1 ( &! ' ( ]! "' ¤ $ m 5_7X! '#")"" ¢ ')( $ ! % - c - f & ' % ' m 5_7X! 8. 7 ( ! ' 7 $ " $ 5n'! % &-

& 3 # ' % ! -;&-21 F G ¤ 9. 1 7 7 $ "'& (]! ' $ 5n'! % & D -;&- F G 3 -;&-!& F G9H = '8> X@CB 10. & &-

&-21 1 "" $ $ 5n'! % &- & #')(:7X! % ¤ 11. 1 $ "" n 5 ' ! ) ' * ( O ! # ) ' (:7X! % ¤ $ % % 12. &-

& 3 1 $ "" $ 5n'! % &-

& ]')( ! % 3 ]')(* ! % #'T(:7! % ¤ 13. 1 " ( ' V7 ( m m" $ " $ 5n'! % &-

& #' % ! N 7( m !V7

@CB

0 ' /

4.

5.

6.

¢ ¢ &- -21 3:3 43 Q - c - 4 Q F $ 5_7 G &'%& ( ' : ( 7 V ( Q : ( 7 V( (*')(:7 ( ! '6_ " " 5 7X! % (P5_7!A 65:'g5 7! D & N L J '( 6(*'! " ¢ &-21 65_7X! " c F $ 5 7 G F $ 5_7 G & ' '& " ' ( 7 ! '65 7! % ( P5n'65_7X! 65 7! ( 5:'g5 7!Ij + c lC- c - - &- . F ¢ ' a 1 $ 5 7! "" ( g5 7X! " 43& c F $ 5 7 G a 4 3 & &!" $ 5 7 ! % ( 6 5 7! " a &-

3 ¢H ¤

G ¤

¢ 4& ¤

"

7.

J L 5_7 c - ¢ Q a 4 3 4 & $ & $ ( 5_9! 7X" ! % F3 G a 4 ¢ 4& ¤ & -

5.11. The Gegenbauer Polynomials Cnλ (z) λ nk

5.11.1. Sums containing Cm 1.

2.

3.

(z)

('! " '! " S 5 ! $&% $ % S - - 0 & ')( S '65 43 ¢ S ( % ! ' ! ' % £ & S - N '6 5_7

1 ! m !" ¢ ' C7 ( 9! " S c - 0& $ L 4 3 J ( (QS S V7 (Q( m ( 5n' 7% ( 6( m ( ! ' 1 ! ( 9% ! (C7C(% 6( ! ' ( Q ( 5n' #7(Q( m (

7 &N

S "1; - ¤

!

" ¤

(9! " (&! ( (P5 ! " c S a 0 & 4 3 ¢ SEa 1 -

1 ( J L£ . a ¢ Y D j SE+ -!a &4 l c &-21 3

'I@ B & ( J L ( 9! " ( P5 5_7! " c SEa a &

1 &! ( & P5 -21 D (P(5n '! ( & c SEa a & ?7R@CB

( 9! ' & $ 5 9!&'! " ¢ ')( $ ! % (Q(*'65_7X! " (P5n'65_7X! " c 0& ' % ( &! ' & ¤ 4 3 $ 5 P5_7X! & L" ¢ J

43 ')( $ ! % 5 1 L " ( P5n'65nO! " c a 1 & J

£ d2 ( 5 7! ' & ¢ 3h ¤ ' % 5 1L' J

( ( 9 ! C 7 ( * ( 9 !" " ¢ ' £) £! ')( (P5_7X! " S c & $ L 4 3 3 d 3 -

J 3 £ 3 £ d2 43 3 d2 a ¢ 3h c a & = > O'I@B S - 1 Q(*'! " C7 (* Q(*'! " 43 ¢ $ L0d 3 £) ')( $ ! % ( (* c 0& ') ( P5_7X! " &-

J 3 £ S 3 £ 0d2 4ST3Pd 3 4SEa ¢ 3h

S c a2S & = 'I@CB 1 &- S

λ 5.11.3. Sums containing Cm

1.

pk (z)

nk pk (z)

")" ' & -21 ')( $ ! ')( $ ! % C7 (& ! " 63 ¢ - c - 0& ( ('! " Q(*'! " (:7 ¤ ' J % C 1 7( L &' ! F 7(8 G F G ' (*'65 1 Q " L J

3 :;

.

; 3

7 $ ')( $ ! C7 (*&! " ')( $ ! % C 7(%9! " F 7(8 G c - & F 3 d G ( 7 ) : 1

2.

3.

4.

5.

6.

7.

8.

m !" C7 (& ! " $9% )" " C7 ( &! "

,

= '*>?7R@CB

( ' m (

7! ' 1 ! £ - ¢ 3h& - c - & m 5_ ¤ ' % &N

m 5 7 # 7( 1 -

¢ 3h& - c - & 7! % . 5 2 CP7 5_(87 ! D ¢ 3 ('65_ &! '%& j + -a c -21Rl c -;&-!&4 F E(:7 GIH 1 (9! " S a2S $ 5 ! % c a 0& &"( % C7 (&! ( S 43 ¢ $ L c - S F $ G SEa J ' (9! " $J L P5 ! " £ 0 ¢ S c a - 0& 7 S 9! ( F 5_ a &-21 lC- c - S -; . F 7 ' !

> @CB

1L " 3 ¢ S a - 1 & L $ 5 J ! % ( 1 L)" O' ' J%

7 . ' L D 5n'! % j S + 9lC-;&-21 £ 3 ¢ _'I@CB J ( 9! " S a2S $ 5 ! % c a &

&-

"( S % C7 ¢ $ L ¢ 3 SEa c - S F G ¤ ( 9! ( 3

SEa 7( $ J S 5n')(* $ " ( &! " ¢ ( $ L $&% &! " 3h

SEc a a & J &-

5n9! ' ' > @CB &! ' S c 0& c & = * m !" ¢ ' C7 ( &! " c - a S & $ 4 3 L

J 3 ¢ S ( m 5_5n7X'! ' ! % ( 9! ( N ( (*( ' m ( ( ' # 5 ¤ m &

1

! 3 : L

26.

27.

28.

29.

30.

31.

32.

33.

34.

0 ' /

¢ ' J 5 1L " C 7(%9 ! " c - a S 0& $ 4 3 L

J 3 ¢ SEa ( 9! ( j + -;&-21 l c -21 . ¢ 3 £

¤ SEa 1L( J

¢ ' ( P5n 5n'! " C7(&! " c - a S & c SEa & ¤ $ L 4 3 J

m !" ' C7% ( 9! " 0 3 ¢ - c - a S & $ L

J ( # (*' m ( 1 (Q( $ C 7 ( 5 ! ( & ! ( ' m ¤ W 5n'! % 3 ¢ S & N

m ( (*' 1 ! W

-Q1 J & L " . C7( &! " c - a & £ d2 j + &4 l c -;&-!&4 ¢ 3 £

¤

1 ('])" "'& C7 ( &! " c - a &

1 ( ' ( 1 (*' 9BB B 9! ' & £ 43 ¢ ( '6 a 5_7! % & 1 SEa 1 N S '( 6(*' #7 &B BB #7 " ¤ ! " " ' & ;5_7X! c -;a ;5 7 , ¤ ')( $ ! % C7 ( &! " c - a 0& C7 ( &! '

1 ) *

1 I 1 ( L' 7 ¤ ¢ ')( $ ! % C7 ( &! " 0 3 - c - a 0& ' J% & L ' ) (87 ,

1 J

J&L" ¢ C7 ( &! " 3h

- c - a &

1 3 ¢ £ d2; ¢ 3h O -; j + c -;&-21 l &4 . £ 3 ¢ ¤ ')( $ ! ' &-21 ¢ ')( $ ! % C7 ( &! " c - a 0& 43

1 ( '! " ( 1 (*' L " #( &! '%& a 1 3 ¢ J (* '! " - = '*>?7R@CB '6 ( ' % & L ' J

3 : P

.

35.

36.

37.

38.

; 3

7 ¢ ' ' 2- 1 ')( $ ! % C 7 (&! " c - a & $ $ L L 4 3 J J

1 £ d2 O' '% ! % N ( ' 7 ( ' P 5_7 ¤ &

& !

' &L " C7(%9! " J ' ( $ 5_7! 3 ¢ - c - a & 'g5 7 0 3 ¢ -;&-21 $ L J

1 . ' L ¢ £ ¢ Y '( J 9 ! ' & c -;a &-21 0& 3 43 j + c a -;&!- &4 lC-;&-21 3 ¤ & 1 m !" ¢ ')( $ ! % ! " C7 (9! " c - a 0& 4 3

1 # 5 7 £ d2 ' % ( m ! ! ' N ( ' ( m (* ' 5_7 ! & ¤ ' & m 6

&L " ¢ ')( $ ! % J m ! " C7(&! " c - a 0& 4 3

1 9! j + c a !- &4 lC- -; a &4 . £ 3 ¢ ¤ m '

'65 7! % $&% C7(&! " c - a & -21 D ¢ 3 C7 (&! '%& c -;a & -21 0& H ¤

1 ( 1 (*' L " J ')( $ ! % C7 ( &! " ( Q(*' ! " - c - a & 40.

1 ( #$ L' 1;- P5_7 1 (% ¤ £ d ' % ( J & 5_ 7X! ' 1A- N &

& &

! ( L" &L" ( P 5_7 $ 1 J ( Q(:7X!A $ (*P5 ! ')( $ ! % ' % 41. ( 5 &J L " ( &! "'& $ J 1 ( L '% & Y ¢ 3h - c - a & 3 ' % ')(* Q(:J 7 !A ')(* 5 !A '( 9! ' ¢ 3h -;

1 (' #7 # Q(*'65_7 7 (8 ¤ Y N ! & c a 1 (*' c a (*' 39.

3 'I@CB & ( J L 5n'! %

3 ' ! > @CB O' 1L " ¢ a 1 0& ' ( $ L $ 5 ! J% ( (

3 J

S - a 1 1L" 1 7 O' J ' L D ' 5n% '! % j + 9 lM1 -; . £ 3 ¢ 'I@CB S

1 J ')(&! " (& ! '%! & 0& ¤ 3 £) d2 m $9(% C97 ! ( " ((* ' ! c 0 & " m m ' &-

WX\ X\]W . (:7 5n ' & 5n')( $ G &c !- & 0& ] d j + l F G ¤ $9% F3 3 ¢ ')( $&% $ ! % d2 7(* c a 0& ,

)

&- (' V(')( 1 V('65 1 #$ &! ' F G N ¤ (

' 8 ( 7 # 5 1 (* "

& 1 J L'

! 43 ¢ ')( $&% $ ! % d2 7(8 c a & ,

)

&- a 1 " ' (' V(')( 1 V(')( 1 #$ 9! '%& a 1 N " ¤ & ( O')(:7# 5 1 ! (*

&L ' J

$ 5n Q(:7X!A m ! " c -Q1 " ( &! " J 5 L (*' ¢ 3h

c a 0& a ( : 5 g ' & $ %

&-

c & m L" 1L " 1 L " J J J

( ' ]'65 #$

c a 1 ( m Q(:O'7X!! % ' & N ¤ & c a 1 # 6(* m 5 1 ! 7(*

$ 5n 6(87!A m ! " c -Q1 L " ( 9! " ¢ 3h c a 0& J5 ( : 5 6 ' ( ( ' a 9 $ %

&- a 1

c & mL " 1L " 1 L " J J

J

( ' ]'65 P5_7 #$

c a 1 ( m ¤ Q (:'675_! 7'%! & % N & c a 1 Q(* m 5 1 ! 7(*

3< ,

51.

52.

53.

54.

55.

56.

57.

0 ' /

5 7 $ (:7X! c -Q1 L " (9! " a ¢ h & 3 ¢ J & c & 3

5 '65 1 L " 1 (*' L " 9$ % Pn

&J

J

(' 5 ' #$

c -Q1 # : 6(8 '7!! % '%& N W ¤ a & c -Q1 c 1 ! &- &

58.

59.

5 7 $ (:7X! c & 3 ¢ 5n'65 1 J $9% P L " J

J

-Q1 L " (9! " ¢ 3

& c &a ( 1 (*' L " h

(' 6(87! '%&

c '65_7! % & N

c -Q1

λ

5.11.4. Sums containing Cm 1.

2.

3.

4.

nk pk (z)

a 1 & #$ -Q1 # 5:'65_7 W ¤

c a 1 &- &

!

and special functions

']" " (:7! ')(* $ ! % C7 (&! " #& - &- c - 0&

]! " ' 3 ]C7 (% 9! '#" c - a 1 & = '*>?7R@CB

& -21 " " " ' ]! ]! ')(* $ ! % '( 9! "'& &- c - a 0& 3 # '( 9! ' c - a 1 & ¤

1 m 5_7! ' m !" C7 ( &! " fI c - & ' %

7 Y D 0f$ ¢
$ 5 !A $ 5_7! % ¢ 3

' 5 L " ( 1 (*' L " h 6 J J Y a a &

&- 1

7 (

N & 7

*

N &

j a 1 ) * ¢ X £ X £ Y

N &

7 @CB $ 4 3 L l J 1 ¢ ' ¢ S £ d2 S S 3 9! % ( S S c F $$ 5n * ( G $ L 4 3 h 3 & 3 S l J 1 = '*> @CB (9! ( % ( &! ( ¢ ' c S / 43 ¢ S ! % £ & 4 S S l 3 43 ¢ S % $ L 4 3

J 1 = '*> @CB % ( 9! ( & ¢ ' c SEa / 3 ¢ S 5_7! % £ & SEa 1 S l $ L

4 3 2 1

1 J #( &! ( & '8> @ B 1 3 43 ¢ S = % ( 9! ( ¢ ' S '8> @CB c S F $ G d2 S S l 3 ! % £ & S $ L 4 3 =

J 1 3 L1

.

¢ ' ES a c SEa F $ G £ 0d2 SEa $ L 4 3

1

1 J 1 3 % ¢ ' c S / ¢ ! % 0d2 S $ L 10. 4 3

J 1 ( 7! " ' c SEa / ¢ 3 ¢ S $ L 11. , 7 5 $

1 J 1 9.

; :

1 S l (&! ( & 5_7X! % £ & SEa 1 = '8> @ B &! ( S 43V& S l 3 ! % = '*> @CB ( & % 5_7X! % 0d2 SEa 1 S S l 9! ( & 3 5 7! % = '8> C@ B

¢ ' $ L S c F G 3 (& ! ( ! % £ & S

S $ J S -; (P5 ! ( "' £ S O')(* ! " (* '! % ( P5_7! " £ & -

& - a $ 5:'! % ')(* = '*> @CB ¢ ' SEa c SEa F $ G $ 4 3 L

1

1 J ( &! ( & ' % ( &! ( & ( P5 5_7! ( "' £ S 1 (* '! % 3 5_7! % £ & SEa 1 & - a 1 S ')(* ! " Y -; a $ 5n'! % 'T(* ( &! " (* ')(* $ 5_7! £ & - = '*> @CB ( 7X! ( % £ ¢ ' S ¢ c S #F " $ G ! % d2 S S l

43 J $L ( 9! ( 1 3 ! % V& S = '8> @B ( ¢ ' $ & ¢ G 43 J $ L $ 5n ( c SEa 1 F " $ 1 ( & £ SEa S '8> @ B ( 7! 5_ 7X! % % £ d2 SEa S 3 ( &! 5_ 7! % 1 =

1 l $ ¢ ' 43 ¢ S % ! % £ d2 S S S $ L 4 3 & c S * $ 5 ,

l

) J ( 9! ( 1 % 3& S = '8> @CB 3

12. 43 ' % (&1 ! (

13.

14.

15.

16.

3 L,

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

0 ' /

$ ¢ ' & SEa 1 c 5 , $ $ L 4 3

2 1 E S a

1) * J 1 3 ¢ S 5 % 7! % £ d2 SEa S 3 #(&! ( % & 3P& S = '*> @CB

1 l ¢ ' d2 S S 3 &! ! ( % S S 5 $ $ 4 3 L & c S , l

) * J 1 = '*> @CB ¢ ' SEa 5 , $ $ L 4 3 & 1 c SEa

1 )+* J &! ( & 1 43 ¢ SEa 1 £ / 0d2 SEa 1 S l 43 ¢ S 5_7! % SEa 1 = '*> @B 7 $ 7 ¢ ' S S c * " $ , 43 ¢ S d2 SQ - S S $ L 4 3

l ) J (9! ( S 1 8 ' 3 % = > @CB &! '%& ¤ $ "" a ¢ £ ) ' : ( 7! % P5_7X! 9 $ % c 0d2 3 & &- & 3 1 m $ m 5 ! '#)" "" ')( $ ! % C7 ( &! " F OG c - ¢ & ' L ' 5 J !A C7 (&! c -; F ¢ 3 G ¤ ' m m " ) " ¢ ')( $ ! % C7 ( &! " f &- -21 c - f & ' 1 ( ' % ' 5 J 5_7X!ACL 7 ' (* &! ¤ ' m ! )" " ¢ ¢ O')( $ ! % C7 ( 9! " f &- -21 c - 4 f 4& " ' O'! % J ' 1 L ' 5_7!AC7 (%9! ¤ ' m (&! ']" $ ']" c a F $ G O')( O! % £ & &-

& $ % = '8>?7A@CB 0 2 d

&-

1 (&! ' $ ']" a '8>?7A@CB c a F $ G ')(:7X! % £ & &-21 9 $ % 2 d =

&- 1 1 3 L 3

.

27.

28.

29.

30.

7X! ¤ 1L " a ¢ 3 '6')5_ J

1 : ( 7X! % G

F

$ n 5 '! % 1 (*' L "

&-

1 J

" 1L " ¢ a J

1

F G , 7 5 $ a 5 n ' ! ( ' ( $ %

&- 1 1 L" 1 J

3 '6')5 (:7X!! % ¤ " 1L " a 1 F ¢ G

$ 5nJ ' ! % * ( '

&-

1 1 J L)" 7 O'65 ! 7! 3 '65_ D ')(:7! % ')(*O! % H ¤ " 1L " J a

$ 5n'! % ( ' ( , 7 5 $

&1 L" 1 J

3 '65 1 J ! % L " (*'

n 5 ' $ 1 1 J '65 L

6 ' _ 5 7 A ! 3 #')(

31.

" !AO'Q5 !%

33.

34.

1

a 1F !

" 1L " a J

1

, 7 5 $ n 5 ' ! ( ( ' $ % 1 L"

&- a 1 J 5 !AO'Q5 ! # '65 3 '65 !A #'6 ') ( !% 7 3 ( 1L " $ " J

(*' a 1 F ¢ 5 $ n 5 ' ! $ % m

&-

1 L" 1 "m J

"m 3 ' % ! 3 '65_7X! % ')(:7! N &

$ 1

¢ G

7 O'65 ! D ')(:7! % ')(*O! % H ¤

a 1 F ¢ G

&-

O! # '65_7X!AO'65 ! '65_7X! ¤ 7 ')( O! % 3 3 ')(87! %

32.

; :

F1 ¢ G !AO'65 O! '65 ! ¤ ')( O! % 3 ')(87! %

G

( ' : ( 7')( m ( m m !

" $ " 1 ¢ a " L 5 m $ 5n'! J % ( * (1 ' L " 7 @CB 1L " $ "'& $ 5n'! J % ( (*' a 1 a F ¢ G , 7 5 $

&- 1 1 L" 1 J

* ( ' ] 6 ' 5 O 6 ' 5 V ! ')(* ! % £ # & N ` & ^ 7 (*' 'g5 ` 3 ')(:7 & N ^ = '*> @CB ( &! " 5 1 " L " £) a a ¢ £! d2 d2 -21 $9% ( P5n'6J 5_7X! " (*' c 1

1 L " &-

" ' J

P5_7X! ' ¤ ' % 1 L ' J

" # ' " ) " " ( ! ( ! $ $ m m ')( $ ! % C7 ( &! " c - F $ G

' ']" 1 ( ! ' m ¤ ( 7! ' % ' m ( !AC7 (9! ' c -; F " G

"$ '& ( m ! " $ m ( ! ']")"" c - a F $ G ')( $ ! % C7 ( &! " ' ']" 9 1 1 # ( 7 ! ( 7! ' m " ' & m ¤ ' % ' m ( !AC7 (9! ' c -;a FO" G

1 " $ 5 ! '#)" "" ( ! m m 'Q( $ ! % C7( 9! " _& c - ) $ 5* , *

' m ' 5 ( !Am C7]!( 9! c -; ¤ ' )+* m ( , m 3 L :

50.

51.

52.

53.

54.

55.

56.

0 ' /

( m ! " $ m 5 ! ]' ")"" ) ' ( $ ! % C7 (& ! " & a 1 c - a 5 , $

1 )+* ' & m " m ( ! ' & ( 7 ! m ¤ c -;a ' m 5 !AC7 (9! ' ( , m

1 ) * ( 7X! " (9! ' O]! ' ¤ ¢ ')( $ ! % C7 ( &! " f &-21 c - F $ m 5_7 G '! % ' m 5_7X! ( 7X! " ¢ ')( $ ! % C7 (&! " f &-21 c - a F $ m 5_7 G

1

9! ' & O]! '%& ¤ ( '6 5_7! % ' m 5_7X!

$ m 5:7! '])" "" 'g( $ ! % C7,( &! " 43& - ¢ f ¢ 4& c

)

7 75 $ m 5:7! , ( L' 1 O'J ! % ' 5n7X! F3 G ¤ m

$ m 5 7 ! ']")"" ')( $ ! % C7 (&! " 3& - ¢ f- 7 Y c - a

1 ) 7 'I@CB & l S J 5 65 5_7X! " (:7 a . '

5 5 7! " $ 5_7! F G j S + l & J $ L '65_7X!A 5 g 5 !AC7(*]! 5n'65_7X! . Y D - j S + -21Rl & 'I@ B &

5_7 m ! 1;-

a . ' . j + l 0& j S + a 9l V-; 0& 46. = > 'I@CB J $L S . ' j S + a l 0& $ L 47. J S F 7 (8 S a . 5 G 43 F (:7G j S + - - - -21Rl -; F (:7;G 1 = > ' ! > @CB 5_7 ¢ ' ¢ a a . 0& $ L 4 3 3 F G j S + l 48. J . 3 3 83 j S + -;a 9l & = > 'I@CB ' 5 7 a a . . G j S + l 0& j S + a 9l 0& $ L = > 'I@CB 49. F J 41.

3 P L

&' 3 465 78

' 5 g5 5_7X! " 5_7 a a . )(*'65_7! " F G j S + l 50. & J $ L ( ( ( ! ! ' j + a 9lV- . & = >_'I@CB ' S 5 65_7! " 1 ( m L "

. ')( J $ ! % C7 ( m ( '! " 5_7X! " £ 63 ¢ j + a l V- & 51. (

5 g5_7X! ' ( O' m (*' V( (* ' 43 ¢ J 1 ' % m L'! 5_7X! ¢ Q3 & N m (*O' V( ( (* ' ¤ ' m ' ! 1;-

5 g5_7X! "

£ - ¢ & - j + a l V- . & 4 3 52. ) ' ( ! ! 5

( * ( ) ' ( $ % m " m L " J ( ! '

5 g _ 5

7X! ' £ ¢ ' % m ! ' m ( ( ( & L ' -; & -; J ( ' ] m ( ( g5n' 5 (* m ( 'g5 ¤ Y N * ( ) ' : ( 7 ( ( ( ' a &

! 1

C( ( ! "

. ' £ ¢ - j + a l V- & 53. S J $L 1 ( (*' L " C( ( ( ! " - $ ( V( (*' ( ( 5 g5J 5_7X! ( 5_7 S

G % F N

( (*O' V( ( (*! 1 a = > '*>?7R@CB 5 g5 5_7X! " ' J 5n'65 1 L "

£ 63 ¢ j + a l - . 0& 5 5_7X! " $ L 54. a2S J 5 g5 5_7 ¤ 5_7X% ! ( N ( (*O' 5_5n7 O'65 5_7 7

&

! 1;-

C( ! " . a j + l - 0& 55. ')( $ ! % C7 ( m (*'! " m ( g5 1 L " 3# ( ' ] m ( g5:' ( m ( 'g5 7 5 g5_7X! ' J 5_7

( 2! '

¤ G ' % m ! ' m ( g5 1 L ' F & N (*'65_7( ( ( ' ! a J 1

( ( ! " m 1 L" . ')( $ ! % J C7 ( m ( '! " ( ( ! " F 5_7G j + - l - 0& 56. ( m L ' ( ! ' 1 43 ¢ ' J % ! ( ( ! F 5_7 G N ( ' (* Om ' ( ' ( O5 '6 5_(*7 'ga 5 7 ¤ ' m ' & m ! 1

3 P P

.

5 g5 5 7! " 5_ 7 ' a l a . & + G j $ L 57. F S ( ' g5 5 7! " J 1 ( * J g5_7! L " ( V( ( ' 5 g5 5 7 ( 43 ¢ S % N ( (*O' g5_7 a 1 = &

!

( 2! " a . +

l j 3 # & 58. ) ' ( ! ! ( ( * ( ' $ % m " & m )" L J

5 65n')(:7 (* m ( ! ' 5 g5_7X! ' F 7 '*>?7R@CB

( (*' ¤ a1

( !

= 1;- > 'I@ B

C7(* m (*'! " 1 L " J 7X! " ( ( 2! " F]3 7,5n G j + - l- . 0& ')( $ ! % C7 ( m (*'! " g5_

, 7 5 ! 5 g _ 5 7 ! ' ' m ' % ! g5_7X! F G ' m ' ( ' 1 ( m (*' ( a 1- - Y N & m 5 1 V( a a 1;- - K- ! . W

+1 a m ! " 1 L " . J +

l- 0& , 7 n 5 G j

] F 3 ')( $ ! % - a 1 L " g5_7X! " ( ( 2! "

J m ! ' ( 2! ' £ ¢ ' % 1;- - L ' ( ( 2! ' 43 -; & -; J

(' a - a 1 a K- 5_7 - Y N W & 7 ( a K- a 1 K- 5 7 ! + 1 a .

C7( m (*'! " . ')( $ ! % 65_7! " ( ( ! " F3 7,5 G j + - l - 0&

( O ' #7( m (*O'

5 )(*'65_7 ! ( ! O'm ! % ' ( ( ' ! F]3 7 'I@ B

&

5:'g5 7 5n'65_7 ! 1;-

5 g5 5_7! " (87 a a . a .

+ l & j + -;9l a -; & j 9 $ % S , ) & 5:' L j + -;9l V-; . 0& ¤ SEa J 5 g5 5_7! " (87 . .

+ a l a & j + a a2S l a -; & j 9 $ % S , ) & 5n' L j + l -; . 0& ¤ SEa J

5.12.5. Sums containing Pm

(ρ pk, σ qk) (ϕ(k, nk

1.

¢ ' . j S + l & ¡ $ 4 3 L J

z))

= 'I@CB 5N;

.

2.

3.

4.

5.

6.

7.

8.

9.

; !, ;

¢ ' $ L j + l . & ¢ F3 G 3 ES a J S 5 g5 5n 'g5 7! " a a a a . ¤ Y a $ 5n'! % F ,G j S + - 9l Q ¢ ' S j + l . F $$ 5n (* G 3 ¢ S ¢ S S l $ L 4 3 h 3 & S J 65_7! ( S '8> @CB 1 3 = % . $m 5 ¢ '$ L 4 3 f & SEa j SE+ a l F $ m 5 G J S ( !" . - ¢ 43fI S F3 mG ( $ 5n + l a a F G ¤ ' ! j % a S "$ " ¢ F3 G j + a l a . ¢ & & $ % &5 g5n'65 7! ¤ 1 3 5nO! ']" # ')

(87! % m . $ m 5 ! '])" "" ')( $ ! % ( ( 2! " F 8G j + - l - ¢ & '

' 5 J !A ( L ( ! j + -;9lV-; . F ¢ 3 G ¤ ' m m ] ' " ) " " . $ m 5 7! ')( $ ! % ( ( 2! " F G j + - l- ¢ f ¢ 4& ' ( ! ' ( L ' % ' 5_7X!AJ ( ( 2! ¤ ' m " ( L . ¢ $ " ¢ J $ 5n '! % j + l

& 5 $ m & 1 "m " (')(:7 g5n' #7 3 ]' % ! 3 '65_7X! m % g5:'! N W 3 ¢ ¤ V ( m m &

!

. ¢ 65n'65_7! " $ 5n'! % F3 ,G j + l O&

& 1 ]7' % ! N (' g5n'65_7 #7 3 ¢ ¤ &

5N :

&' 3 465 78

10.

11.

65n'65_7X! " $ 5:'! % F 3 G

1 5 ' 5n7X! " 8 $ 5*'! % F]3 , G

1

g 5n'65_7X! . j + l ¢

& 3 ')(87! % ¤ &-

. j + l ¢

& & ' 5n7X! 3 Q58' 5:]7!A'g ( 58 ! ' % 58 ! 3 Q 5* 'g(*7X! % ¤

. 65n'65_7X! " $ 5:'! % F3 G j + l ¢

&

&1 65n'65_7!A 65n'65n !A g5n'65 ! ] g5:'g5 7 !A 65n'65nO! 3 #')( !% 7X ')(* ! % 3 g 5n'65_7X! ¤ ')(87! % 3

12.

13.

14.

15.

16.

17.

)£ ! " 65 m 5n'65_7X! " a . ¢ )£ I -21 9$ % m 5n'65_7! " F]3 G j & + - l I & 5 7X! ' ¤ '_ % a . ¢ £! ¢ £) ¢ 65n'65nO! " n 5 6 ' 5 7 ! $ %

F3 G j &+ - 1Rl & !- & '65nO! ' 7 % # 6V5n ')(:7X! % ¤ a . ¢ £! ¢ )£ ¢ 65n'65nO! " 5n'65 7! % F 3 ,G j + 1Rl 4

& 4 - $ & 7 # g5n'65n ! !A g5n'65 ' % 7 ')(:7X! % 65n'6O5n #')( O! % $ 5_7X! "'& g5n'65n ! " a . ¢ £! ¢ 5n'65_7! % F3 G j + 1Rl & $ 5 7! 5 m $ & " ( ' (:7#7 g5:'g5 #'65_7X! g5:'g5 7! ¢ 3 N W % m & 1- 1 a !

. $ ']" a F¢ $ G & $ % 3 g3 j + l &1 5 65n'65_7! ']" 5n'! 3

')(:7! % F G 5N

&-

(')(:7V(')( 5 #$ ! '%& " 1 ¢ ¤ '65_7! % O'65 7!A

5 2! & N 1 3

! W

( 1 (*' L " ¢ &-21 ')J ( $ ! % 5_7! " Y a F $ 5_7 GTj + lV- . F ¢ $ 5_7G

&- 1 (')(:7V(')( 5 #$ '%& " & ! ¢ ¤ '65_7! % O'65 !A

5 2! & N 1 3

W !

¢ 1. 3 &- -21 1 Y j + l V- . F ¢ '65 7 ¢ £ G 3 Q F 2.

; !, :

5N L

4.

$ 5_7! "" 5 7! "

&' 3 465 78

. a a ¢ Q j + l- F ¢ & - ¢ D ! 7 '& . ¢ a ¢ c 4 Q j + l V- F $ & ]! " '6 5_ 7! % 5 ! ¢ 3

$ 5_7 G

7 -2a 1 1 ]F 3 G 3 '65 7! % H ¤ $ 5_7! "" 5 7 ! _ 5 7G 5. " ( ' (87 5 ¤

N ! (

. ¢ $ 5_7! "" a 5 7! " 43 Q c 4 3 Q j + lV- F $ 5_7G 6. & 7X! " F 3 G $ 5_5 7X! g% 5_5_ 7! " c a ¢ Q ¤ & ('6(*'! " . ¢ ¢ 7. &-21 5_7! " c F $ 5 7 GTj + l V- F $ 5_7 G & % ' & " (')(:7V( 6( ' (87

5 43 ¢ '65 7! % ( P5n'65_ ¢ ¤ 7X!A 5 2! & N 1 3 !

. ¢ ¢ &-21 - -; &-21 F $ 5_7Ggj + l - F $ 5_7G 8. &' " '& a ] ( 7X5 ! 1 - a - -;&-21 F]3 !G ¤ 6 ' _ 5 7 ! , G % 3 F 1 ( (*'! " ¢ a . F ¢ $ 5_ 7G &-21 C7(%9! " c - F $ 5_7 GTj + l 9.

& "'#" 65_7! '%& " '%& '65_ 7X! % 5 5

g5_7X! ' 5n'65_7!A( Q(:7X! ( ' (:7# 6(:7V( ( ' (87 W ¤ Y ¢ 3 N ( ( ( ( O')(: 7 ( &

1 !

¢ 5 g5n'65_7! " -21 C7 ( 9! " 10. F3 G Y c - F $ 5_7 GTj + a l a . ¢ ¢

&- ! " ( ')(:7 # Q(:7 5 g5n' _ 5 X 7 ! ' '65_ 7X! % ( Q(:7!A

¢ W 3 &N

¤ 5 g5:'! V( 1 ! (

5N P

.

; !,
X@ B

(' 5 $ I m !5 $ $ 5 7X! $ 5 7! "" m ! " '*( $ ! % $ 5 7 ! % ! " a 1 N !25 $ # ! $ h 5 ` ^

4.

m ' % 3 #'*(?7! % ¤

7

5.

( 'g5 $ m 9! 5 $ ' $ m 5 !" m !" 5m _7 ! " a 1 N !&5 $ $ m _ 5 7X! ` $ $ L J ^ !

6.

( ¢ ' S a N $ L 4 3 1 ^ J 1

( ' & m ! 1 C7( ! #$ ¤ ! a N a 1 ! 1 5_7

m ! ¢ ? S l ! ! ` 3 S m !( 3 ! ( 3P& S

5, ,

=

'I@ B

7.

8.

9.

¢ ' S a N 1 4 3 J $ L 1 ¢ S S 3 -;

' ( $ V m ! ! f O &- -21 a 1 N J $L 43 fI ! 1 ' 3 &-21 ' 5 a N ( ' ! V ( m ! ¤ m 1 !

( $ m ! ' ¢ -21 &- -21 a 1 N ! J $L fg3 ! a 1 ' ( ' 5 7 ! m m '65_7!& (*'! a N ! m 1 ! a (:7! m 5_7! ' ¢ 3 a '65_7! m (:7X!

11.

12.

m !( m ! ¢ E S a S ! ` 3 1 ! ( ! ^ m ! ( "' S -; ¢ a a 1. ! ( " ' a J $ 5n' L43 + ' ( ( 5_7! Y ' ( ( 5 7! " = > 'I@ B " m

(

10.

! 1

( ' m ! (:7 ¤ 1 N !(87 ! a 1

( $ & m ! ' m !' ¤ ¢ ' ¢ f &-21 a 1 N ! ' m 5_7 ! ' 43 J $L ! a 1 ( m ! #$ ¢ ' ; 1 ¢ f 4 &-21 a N ! W 43 J $*L ! a . + 1 ! ' J 1 L 5_' 7 m ' ¤ !' m ( m ! ; 1

a 1 ¢ O '65_7 ¢ f 4 a N ! W 3 J $ L ! a . + 1 5,93 #$

¡\¤

.

; !?7 ! ! ( " ¢ S ] (:7X! % (87! - -21 ¢ ¢ £ a ! 43 % 3 = >?7 ( (

¢ 4 3 S # 1. 1 ( " "" Y 4 3 ¢ S 2- 1 ] ( $ ( ! % 43 ¢

2.

$ ( "

( $" ! ¢ 3 4 3 $ 1 S S - ( 7X! " "'& 3 $ 5 7! % $ 5 7! Y £ 3 £ 3 £) 3 £

!

!

"'& 5 7! % $ 5 7! £ 3 !£ 3 £ C@ B (

SEa &

(

@CB

6.4.4. Series containing products of Si (kx) 1.

2.

( " ( 7( ! " ¢ S 3 # (:7! % $ 1 S - ( 7X! " "'& S - -21 ¢ 3 $ 5_7X! % $ 5_7X! 43 Y ¢ 3 £ a - SEa & £ 3 £ 3 = >?7 ! ( ( 7' ! " 3 7 $ 1 1 1

(:7X!

& 5 7! % _ 5 7! £) 3 £ ( !

!

'

(

@CB

B

6.5. The Fresnel Integrals S(x) and C(x) 6.5.1. Series containing S(ϕ(k)x), C(ϕ(k)x) and algebraic functions 1.

( 7X! " 7 " £ $ $ : E ( 7X!

5;93

h7 ( ( (

3 ¢ S S - -21 43 3 £ =

!

!

!

( " ( O! % ( ! & ¢ ! % 5_7X! ¢ 3 £ a - E S a & >_ ( ( (

"' F G 1

4&

7 1

3 F G 1 1

'

!

!

1.

2.

" ! $&% " - & # -

B

B

'

k, z)

&

¢ ¢ & ¤

7 ! " 1 & )3 ¢ -21 - 3 )3 ¢ X T3h63 ¢ )3 )¢ & ¤ 5;9P

@CB

6.6. The Incomplete Gamma Function γ(ν, z) 6.6.1. Series containing γ(ν

@CB

!

3.

4.

5.

6.

7.

8.

:1 :1 ,

(]! " ¢) $9% ! " & ¢ £ Q3 ¢ £ &

& 1 & 0 ¤

0 &

L J 5 7 !

5 1;- ( 5_7 ¤ 5 7 ( m 5_7 (O

N _ !

! " (87 $9% a & " " " (:7X! ! + -!& . + . - F m ! " ! " ! " ¢ ( 5_7X! " ( 5 7! " a & 43 ; $&% m L J ! - N 5 1 ( 5_( 7m ( ( m

5 ( 5 7 ¤ ! " L ; 1 J _ 5 X 7 !

N 5 7 (65_7 ! ( $&% a &

4.

5.

5< ;

7 3 £G ¤ 5_7 O ¤ 5_! 7

6.

7.

4 3 1

¢ J

J 7 3 F

a 1L & a &L

Z

7 G &

¢ a 0& 4 3

L J 5 7 ! 1 N

.

+

:1 P1 ,

W

!

#$

£

(87 ( 6 _ 5 X 7 ! " L ¢ ! £ J ! ! 4 3 a 0 &

8. " 1N

!" a 1 L " 1 L " !£ ¢ J J

9. 43 $9% 5 1 L" a & L)" a & J J a £ -21 1 J ! & L ( m !" !" L £) J ( 5 7! " a 0&

! - 1 N 1 10. ; $&% m

W !

11.

¤

¤

V+ 2- 1 . F G ¤ m 5 1! ¤ (* m 5_7

" S 2a S $ 5 ! %

3 - S I + -

K

¤

5_7

/ £

a &

( " S S . 43 ¢ $ 9$ I5% ]! ( 3 $ ! % 8 S

; £! & ¤

6.9.2. Series containing Ik+ν (z) and ψ(z) 1.

J( L $9 %

"

fI a 0&

L J 5_7! 0 fI

3 2.

!&

W #$ 7 #7 L J n 5 ! m N & m 5_7 ! n 5 ¤

" J( L $ 5_ 7 ! % fI 0 & 3 fg3 ¢ 1 0& ¢ ¤ fg3 F G - 0& 5?7 (

!

"'&(!& ( X7 ! " L $ 5 &LJ $ 5 5 &L J J

S Ea & £ 3 £ 3 )£ 3 £ ( ( C@ B

!

6.10.5. Series containing Hν (kx), S(kx) and C(kx) 1.

2.

(!& 7X ! " $ 1

( 7! " $ ( & '& 1 ( 7! ( S - -21 J ¢ Y 3

" " !& 3 H 5 &L J =

5

!

!

4.

( @ B

E H "'&(!& ( &( " " S - ( 7! L L 5 1 L J 5 5 1 L 3 * $ 5 & L J $ 5 5 & L J J

J&

a SEa £ £ £) £ ¢ £ 5_7X! % 5 ! 3 - & 3 3 3 = >?7 ! ( ! ( ( @CB " " '& (!& 7X ! " H 3 5 &L $ 1 J 5 (X@ B =

3.

!

( 7X! " ( & " H $ ( & " 1 ( ( 7!

S - ( L J 5 1 5 5 1 3 * $ 5 L L J J

J

5L95

!

"'& '& L J

&L $ 5 5 &L

J

7X! "

Y

H

L

& S - 2- 1 ¢ ! % _ 5 7! ¢ 3 £ a - ES a & £ 3 £ 3 )£ 3 £ 3 = >?7 ! ( ! ( (

@CB

6.10.6. Series containing Hν (ϕ(k)x) and Jµ (kx)

( 7X! "

1.

2.

(& !& $ 1

H

'&

L L 3 J 6 5_7XJ ! 5 & L J

= 5

!

!

65 !

4.

!

!

6.10.7. Series containing product of Hν (kx) 1.

(X@ B

7 $ 5 X7 ! Q a 1 Q H £! ¢ & ! " Z £ ¤ 5 1 L 3 2

-21 1

J

7 $ 5_7X! a Q H 4 £! ¢ & 1 O]! " Z ¤ £ 5 1 L 0 32

-21 H

J

( 7X! " H $ ( && " ( & " "'& '& 1 ( ( 7! ( 7X! " L S L L

J 5_7J !

$ 5 &LJ $ 5 5 &L 5 1 L 6 5 5 1 L 3 J J J &J

S - -21 ¢ L ¢ £ a SEa £ £ 3 £) 3 £ Y J 3 % 565_7X! 3 - & 3 = >?7 ( ( ( @CB

3.

( 7! " $ ( && H H 1 ( 7X! ( " ( &" !& S - ( L L J J $ 5 5 1 L 5 & L 5 g5 1 L 3 J J

J J 5L;

"'&& L J

& L $ 5 g5 & L

J

7X! "

+

S Y - 2- 1 43 ¢ Y £ 3 £ 3 £! J 3 ( ''7& ! " H $ 1 1

L 5 J & L 5 5 £ = J h > 7 ( ! " ' " 3 ' ' 5 J

2.

&(!&

:1

¢ 3 £ a - E S a & &L (

!

a 1 &L ' 1

!

(

!

@CB

( 7 B

6.11. The Legendre Polynomials Pn (z) 6.11.1. Series containing Pnk+m (z) 1.

2.

3.

1

$ _ 5 7 5 7X!;j $ $ _

7 (

=( 7

3 ¢

7 = @ @ B !

$ _ 5 7 7?7 !

!

6.14. The Laguerre Polynomials Lλn (z) λ lk 6.14.1. Series containing Lnk+m (z) 1.

2.

3.

4.

" (P5n'65_7X! " c $ m 5 !( (P5 5_7! " (P5_7X! ( S S %

' a 0& 5: '

&c -21 &

Sc a 0 &

= '8>?7A@CB

( 5 _ 5 7 m % 5 7 7 8 ( ` $J L F G ( 5_X7 ! 1 N 1 ^ 5 _ ! = R7 @CB ('6(*'! " $ 5n'! % c - & d ¢ 3 ( 6( O'! " a Y ¢ 3 c a c a 0Q3 & 3 43 & -21 $&% c &- & = R7 @CB " (P5_7X! " c & d ¢ X & - c c £ / ¤

5P;

+

5.

6.

7.

8.

" ( P5n ! " c a 0& ¤ m !" m! ¤ m 5_7! " ( 5 7! " c a & 1 N 1 5_ 7L J m !" !" a V & c &

N m m 5 5 1 L " ( 5 7! " J

$ 5:' ¢ 3 - c -;&-21 A + -21 . $ L 0 & c a J

:1 5) ,

m 5 5 1 ¤

! ( G c F 7

=

" £ 3h& ] 9 $ % & 9. P5n'65_7 #Q( m 5n'65n $ 5n'! % a m ! " c a & d ¢ N

5 7m 9 $ % L 10. J ! (P 5_7! ' m (*' ')(* m m ! " J 1 ( m 5n' L " ( P 5n'65_7! " #& c aa 0& ' % N '65 7 # 5 7 9 $ % 11. J !

7A@CB

= @ B

¤

5_7 ¤ L

lk 6.14.2. Series containing Lλ nk+m (z) and special functions

1.

2.

3.

4.

5.

7 a 1.

¢ ! ¢ £ ¤ + c (P5_7X! " d Q c 0& 0d F G c a 1 / Q 7 ( 1 (P5_7X! " Q c & 1 N P5_! 7 5_7 L ¤ J 7 ( P5_7X! ' ( ' V7 $&% & c a & ' % N J P5_7 ! 5_7 L ¤ " W #$ Q J L L a J 5_7X! N 5 7 5_7 ¤ ( P5_ 7X! " c 0&

a Q

)£ 3P& - a Q 3 2 d ; 4 P 3 2 d Q c - 0& c W / #$ # 7 ( L c c ; 1 3 65_7! J C 7 (%6(! N g5_7#7(%6! ( ¤

5P :

# . '/

6.

j Q &

-21 + -

.

A )

4 7 (

,

5

7(

!

7B

lk 6.14.3. Series containing products of Lλ nk+m (z)

1.

2.

3.

" " a S a 3 a 9 $ % 9 $ % Q 0 & Sc Sc $&% ( P5_7X! " c Q c 0& R 5 ! !" / d ¢ 7 ( 7 ( G D (87 H c F

c 0& - c Z Z c #- - £ / £ c c /

a Q3 &-

=

£ 9 /

'I@CB

7 = @ #B #C7 !C@MB !

=

=

7 = @ @CB !

6.14.4. Series containing products of Lλ n (kx) 1.

$

!

c 43 c 1 ( " ('! ( " (P5n'65_7X! ( " 3 (P' 5_! 7 ! ' # ! (:7! % ] (:7X! % (P5_7X! ( ( 5 J & 7L! ( ( " " " % ( (P5_7! ' S - " ( '! " ( 5:'g5 7! " " "" ¢ S ( P5_7! " ( 5 7! " 43 - -21 # (* $ ( ! % ' % ! $9% S - -21 ¢ £ £ £) £ ! 3 43 % 3 3 = >?7 ! ! ( @CB

$ ( "

lk 6.14.5. Series containing Lλ nk+m (ϕ(k, z))

1.

" " ¢ 5 7 c - 4 4& (P5_7X! $

5P
?7

8.

$

!

!

3 ¢

!

¢ S

!

( " S ] ! (87 ! % ( " "" -21 ] (* $ (

£ 3 £ 3 £! 3 £ !

!%

m

!

1 !

m ! (E $ 5 m ! ( 7 ! " ! !& ! N a ` 1^ $ 1 m ! ( m m 3 7 N a ! 1 ^ !& ! `

; ,9P

!

m 9! 5 7 ( m N a 1 ^ !& &! ! 5_7 `

( 7 !

( !

#$

m ! (E $ 5 m ! 7 m ! ( m ¢ ! !& ! ` 3 N a 1 ^ ! !& ! ` 4 3 N a 1 ^ h7 h7 1 !

10.

(

m ! ( m m ! K( $ 5 m ! 7 ! !& ! N a ` 3 N a 1 ^ ! !& ! ` 1^ 1 !( 7 m !( 1 = !& !C@ m 1! ( " !& !( N a 1 !& !( 1 = m !C@

h7 h7

9.

m !( " !& ! ( "

B B

B

B

+

:1 !?7R@CB & 2 1 ! 3 3 &-21 F 3 G

N & 7 (*' V7 ( (*'65_7 a 1 & = ')( $ 'g5 $ ! % a 1 & 43 ¢ * 3 ¢ $&% ')( $ ! % £ & - FO$ G ¤ ¢ 'g5 $ ! % £ 'g5 $ ¤ ) ' ( ! G FO63 & $ % $ % & 3 & * -;&-21

S . & -;&-21 F G + S -21 . + -21 3 ¢ a $ L &-21

J S ' %

W U a [ ¢ ¤ a Y & % B B B ( % 3 0 & Z Z Z

-21 [, 1 a U a ZZZa 1

1 £ & £ -;&-21 & 1 ' $ L& &-21

-21 & &- -21 & J 3?3 ¢ 1 - 0& -; a 1 0& ¤

a & ; F G N & 3 ¢ &-21 F G &-21

;93 P

#

?7R@CB & 2 1 ! 3 &-21 F G

N & 7 (*' #7( ( 'g5 7 a 1 & = a 1 & " O 43 £ & -; - - &-21 £ & ¤ &-21 & . S ' % -;&-21 IF G + S -21

% B BB ( % -21 & ¤ , [ U a ZZZa 1 £ & 63 £ -; & 1 43 ¢ ' $L ¤ & 0 &

&-21

< 1 2 1 J ¤ £ -; F G 1 ' $L 0 & 0 & 2- 1

&- -21

J ;!5)

24.

25.

#

?7R@CB

( ( ; 1 ! F 3 G N & ( ' V ( (*'65_7 H 0& #7( ( ; 1 & 2 1 ! F3 G

N & 7 (*' V7 ( (*'65_7 H a 1 0& ( 1;- ( (*'65_7! " '8>?7A@CB ! 5 $ (*'65 L N & ( $ ]')( $ ( ( 'g5 7 = J

¤ 7 & -21 ¢ 1 L "

& -21 J , G

6. L & L & -; 43 1 L '#)" " F J

']" 7 L F G F G ¤ J ' % & 0 & 3 4 3 7. L a 1

-;&-21

¤ 8. L -;&-21 & a 1 0& ( ; 1 ; F 3 G N & (' V7 ( ! (*' L 0& 9. L a & V7 ( ; 1 ¢ & 2 1 &-21 F 3 G N & 7 (*' 5_7 #7 ( ! ( ' L -21 0&

!& '#" ( 1;- C7 ( (*'! " L & 2 1 G N & ( $ ( $ 5n ' #7! ( (*' 3 J ( $ 5:'g5 1 L F J

= '*>?7R@CB ( ; 1 10. L 3 ; F]3 G N & (' ( (*! '65_7 L 0& -; 0& #7 ( ; 1 ¢ & 2 1 3 ; &-21 F]3 G N & 7 (*' #7( ( ! 'g5 7 L a 1 &

"'%& ( 1;- (*'65_7! " L & 2 1 '8>?7A@CB ! J 5 $ (*'65 L N & ( $ ]')( $ ( ( 'g5 7 = J

5. H ; - 0& 4 3 3 ¢ 3 &-21 "'%& L & -21 J

7.2.6. The Anger Jν (z) and Weber Eν (z) functions

¤ - 0 & J 43& ¤ J 0& 0&

1. J 2.

;!53

#

?7R@CB 3 E 43& ¤ 5. E 0& - . 7 # ' " 7 +

& 2 1 ')(:7! % % G G G 3 H 0& ¤ 6. E 0& F F 3 F]3 . 7 '#" 7 +

& 2 1 ¢ O ) ' : ( X 7 ! G G G % % F F 3 F3 43 7. E & -; 3 43 ¢ H-; & ¤ ¢ ¢ ¤ 8. E a 1 0& 43 ¢ J -;&-21 & 3 ¢ J a 1 43& ¤ 9. E a -; 1 0& 43 J&-21 & 3 J -; a 1 43& #7( ( ; 1 ¢ & 2 1 ! 10. E a & &-21 F G

N & 7 (*' 5_7# 7(*')( E 0& 7 ( ( & £ 3 &- F G &- N & (*' 5n #7! ( ' ( E a 1 0& 7 & - ¢ ¢ a ¢ 3 )3 43 ( ( Y F]3 G a 1 N ( $ ( $ 1;5n- ' #! 7(*')( = '8> X@CB

& - ¢ F G 3. J a 0& & -21 3 - £ &- F G & ! & - ¢ ¢ 43 3 )3

;!5 5

( ( 1 ! G N & ( ' ( (*'65_7 E & 11. #7( ( 1 ; & 2 1 ! G

N & 7 (*' V7 ( (*'65_7 E a 1 0& a 43 ¢ a )3 ¢ F G 1 ( ( Y N ( $ ]')( $ 1( - ! (*'65_7 = '*>?7R@CB

& ( 7X! ' & -21 ¢ O')( $ 5 a G F G 1 12. J a 1 0& 3 F 7 ' -21 $ (:7 S 3 J $ L F G -21 J L F G S F G S - -21 Y &S 1 0& 43 ¢ a -21

a -; &-21 & &- a 1 0& ¢ ¢ S 3 3 1 - S 0& 43 ¢ -;&-21 & ?7A@CB

+ 5.

6. 7. 8. 9. 10.

11.

12.

13. 14. 15.

16.

17.

18. 19. 20. 21. 22.

' c ' % 1 J L' 0&

#

7 ¤ c & . j + -21 lC-21 0& ¤

£ ¢ ¤ 3 3 ¢ F ¢ h 3 G ¤

£ 0& 3 ¢!¤ 43 ¢ a 1 7 7 ¢ £ * 3h 1 , ) ' (:7 ¤ % '65 1 L ' 1 - ) , J

' % j + -21 lC- . F 3 ¢ G ¤ 1 J L' 7?7A@CB

( 7X! ' a F ¢ 3h G ¤ & 7 8 (

1 3 ' % O'65_7X! a 1 0 3 ¢ -21 -21 - ¢ 3 a 'g5 1 L '%&

1 ) * J

' % j + 1 lC-21 . £ 3 ¢ ¤ 1 J L' O' '% ! ! % £ & j + 1 lC- &-21 . F 3 ¢ G ¤ £ £ ¢ ¤

a 1 & 3 ¢ 43 ¢ 3h G ¤ 7 (8 a 1 F 3 #'65 7! % ¢ a '65 ! '65 & L' 0 3 -21 -! &4a 1 - ) * J

' 6 ' _ 5 X 7 ! % % '65_7! % £ & a 1 j + 1 lC- &- . F 3 ¢ G ¤ 7 ¢ & 7 8 ( 3 a 1 0& ¤ ¢ 0& a F ¢ 3h G a 0& F 3 G

1

1

&-21 3 ¢ ¢

23.

24.

25.

26. 27. 28.

29.

30. 31. 32.

2. 3. 4.

7 , ¤

¢ 3 7 ¤ ,

3h

= '8>?7A@ B

¢ 0& 3 43& ¤ 0& 3 ¢ £ -21 0

¤

a 1 & 3 ¢ £ a 1 1 ¤

¤4¤¤ ]S 0

1

[ X@CB

23.

25.

?7A@ B

33.

1 -; & F ,G 1 -; F ¢ 3 G ¤

34.

a -;&a -21 0& 3 F G 1 -;&-21 F ¢ 3 G ¤

1

35.

&L ' £ ¢ a

J

7X! % @ 3 1 a 1 ) -;&-21 0&

&-21 = '65_

36.

37.

c ¤4¤¤ ]S 1 U

[?7A@CB + C l ' 5 ! j % 18. j + lC- - -; & = S -21 n 5 ' ! % S .S ' % 5n'65_7X! ( F 5_7G j + lC- S . & ¤ 19. j + l & SEa . 5_7X! ' (:7 ¤ ' % F G 20. j + -;9l & A. j 0& ¤ 21. j + l 0&

; ;9:

22. 23. 24.

25.

26.

27. 28. 29.

30. 31. 32. 33.

34.

35.

36.

37.

7 . j + Cl -21 0& j & -21 & j 7 . j + lM1 0& 7 (8 j 0& 3 j . j + lM1 0& F 5_7 G 1 j a 5_7 . j + lC-21 0& j * , )

& 0 a 1 0 & ¤ _ 5 7 ¤ 1) * , ¤

&L ' . j + 1 lM1 0 & '6J 5_7X! % 0& ¤ 1L' . j + -21 lC-21 0& J ' % & ¤ . . j + -21 lM1 0& 43 ¢ j + 1 lC-21 3P& ¤ 7(8 ¤ 43 ¢ J 1' L % '

) * , 5 7 J '1 L % ' F 5_ 7 G 1 a 1 * , ¤ ) 5_7 5 . j + lC-;&-21 & F G j 5_7 , ¤ ) _ 5 7 ( . j + lC- &-21 0& 43 ¢ F G j F 5_7G ¤ 5_7 . j + lC- &-21 0& F G j * 5_7 , ¤ ) 5_7 a . j + lC- &!- &4 0& F G 1 j a 1 * 5_7 , ¤ ) '! % 5_7 . j + 1 lC- &-21 0& ' % ! F G ) * 5_7 , ¤ ((:7X! '%& (:7 . £ j + 1 lC- &!- &4 0& 43 -; C7(8! j a 1 ) * 5_7 , ¤ '65_7X! % 5 7 a . j + 1 lC- &- 0& ' % '65_7X! % F G 1 a 1 ) * 5_7 , ¤ ; ;
?7A@CB

3. 4. 5. 6. 7. 8.

4 3 ¢ A F ¢ G ¤ a A &

1

' ( 7X! a F ¢ G ¤ A & 7?7A@CB 7 ( '

&-21 / H & !

( ' (*' #$ 1 0Q3 ¢ ¤ : ( 7

1 ,

)+*

! (' ( ' 7 1 ¢ j 7 3h& F 7 (8 G ¤ ! ( ' (*' #$ 1 'g7 5 7 -21 0Q3 ¢ a 1 a ¤ E : ( 7

& ,

1 ) *

! ( ' ( ' ( #$ 1 ¤ ¢ 8 ( 7 0 Q 3

,

)+* 1

! ( ' ( ' ( #$ 7 1 ¢ a 1 a ] 6 ' _ 5 X 7 ! 8 ( 7 , ¤

6 3 2 1

1 )+* & !

( ' (*' #$ 1 O'! % F G j F 7 G ¤

1 ( O ' ! 1L ' J

( ' ( ' ( #$ O'65_7X! % a 7 1 £ 1 j a 1 F G ¤ - ( 1 (*O'

& ! J L '

(' m ( 7! ' ' % O(:7 . !a + 1A- -; F (8 G ¤

h 3 & - 1 1;- -

J L' ;

60.

N 1

61.

N 1

62.

N 1

( ' (' ( 7! ' ' % 3h& j F (1 *' 1 !

J L' (' V(')( 7 1 ( O' F 3 G ) * ¢ 3 , ! ( 1 1 ¢ 1

E 7 / 5_7 !

#$ 1 1

N 1 &! ( 7 Y K

63.

E(:7 ¤ (8 G ¤

¤

J & L / / ¢ -21 " 7 " g5 5_7! K 3 65 _ 5 7V!

#$

N 1 1!

Y £ 1 ¢ / 1 -21

C 7,5 ! K 5hC7,5 ! C7 ( ! £ ¢ 1 3 / 1 -21 K 5hC7 ( !

65.

66.

¤

F G F G

1 &

64.

¤

1 &

N 1 7! 7 ( 7(* ¤

2 1 £ ¢ ¢ / " 3 / 3h , K 5 7( 7(8 ) L J 1 &

N 1 7 ! ( 7 , 7 5 _ 5 7 1 ¤ £ £ / ¢ 2- 1 K 3 £ 2- 1 ) 5n5n 5_7 ,

67.

1

N 1 &!

#$

&L ¢ J / -21 7 " Y K 6 5_7!

; < ;

K

" ¤ 3 65_7X! 7

68.

69.

L1

7 7 2 IF G 1 K , K 3 , ) * ) * " 7h7A@CB G 9 G H L D F F 3 = J # m - - - $ £ - -21 £ f / £ 3hf ¤ 100. N

1 a ?7 ! = @ B !C@CB ' (' V( 'g5 #$ ( ! O')(87! ' 7 7 7 < G F]3 G H 105. N ( 1 L ' 1 L ' DXF]3

1 1! ( & J & J& & '= 8>?7 = @ B O!C@CB ! 96.

; ! = @ B 7 !C@CB

&A -21

l & 1 J L' #$ ' 1 ¢ @ B 7 C! @MB 43 '65_7 ==

( ' ( ' ( (

N 1 !

108.

&

( ' (87 V(')(

N 1 ! (

109.

N

111.

N

&

!

= = @ B 7O!C@MB

'& a 1 '65 !A '65 !

7

= = @ B 7 !C@MB

G 2 IF G F ?7 = @ @CB

3 N 1

!

5 1 m 1 #$ !

N 1 ( m 5n '65 &

F G S C7 !% ( ( m Y

126.

1 ( m! 127. N 5 '65

1 ( m n ( m 5n'65_7! " Y

#$

&

m ( ( J

a a . 7 1 F G #$ 1m! 1 ¤ N 1 ( m 5 $ 5n 6 ' 5 &

( " ( S ( 5n '6( 5_! 7X! ( $ L 3

m J #$ m 1 ( '! ( ")" ( m 5n'65 & L)" 1 ¤ ! J

N 1 ( m 5 $ 5n 6 ' 5 & m 5:'g5 & L "

1 1 N 1 &

! 1 #$

1

#$

" F 7 G

128.

1

S " " j + -2 1 lC-!& - SEa $9% S 5n'65_7X! " ( m 5n'65 & L" ( m 5:J 'g5 & L "

J

= = @ @CB

129.

N 1

&1 !

= = @ @CB

m ! 1 #$ a J

N 1 a a - J

7 Y a a 1 J L

130.

a L

£ - 0fI $ L 4 3 J

L' J

a - L

;9L,

'

7

¤ a L a - _ 5 7L J

J

131.

m ')( m N 1 1 £ & - / !

Y a J

! ( m ! ' 7 a - L J

'

$J ,L 3 £ - f K3

- a - a 1L

7 a a - a 1L - a J J

! 7 . / m 5 ! ' $9% j &+ - a l - -; ¡ 7 7 a a - a 1L - - 1 L J J J

m ( 5n'! ( ! ' ( ! ' m (* ! (

¤

- L

m V (')( m £ 132. N 1

1 !

Y a - L J

m m (* 5 133. N

1 m ( 5n' ! 1

' Y £ - $ L& fI fg3 £ O 3hfT3 & J S £ 0fg3 £ ' S Y L 0f J 7 7 Y a a ( a a a a a a ( a a 1 L 1L L J J J J

m m (* (*' 134. N m ( ! 1

1 " "

/ m ! ( ! $&% fI fT3 £ O j + a -21Rl - a -; . ¡ ' & 7 7 Y a a a ( a a a ( 1L L 1 L J

J

J

J

5 7 '65_7 3 ¢ -21 £ SEa a 1 F 3 ¢ G 135. N 5

1 ! 1 S ( ! " 3 ¢ G j + l a a - S -;&- . ¡ Y % F 3 S ( O! )" " a a a . ¢ Y -;&- ¡ & $ % 3 j + l &

¤

L

¤

L

;9L 3

L

¤

Y

2- 1 3 £

D £ ¢ H

a a - ( 7! 7 &B BB 7 &BB B ')(:7 P5 3 = ! ! ( #$ 7 1 1 " ¤ G G G I G H F F F D F 136. N 7

1 ! 1

#$ 1 1 1 1 ( 5n J L J & L 137. N &

1 1! 1 1L 1 L J J

1 #$ 1 1 1

g(* 6( J L J L J & L 138. N &

1 &! 1 1& J L

#$ 7?7A@ B = , 7 5 7 8 ( 7 ( 7 8 ( , , ) )

8.

: N,

( ' ( 1 ( ' ( 1 ( ' 1 (*' #$ 15. N ( 1 (*O ' V( ' 1 & ! £ - R&2- 1 a 1 a a 7 ( 7 8 ( < 7 5 7 (8 , ¤ ,

1 ) *

1 ) * #$ ( ' % 1- m m 5 1 ] C 7 * ( m !' 16. N

1 1 &

& !

( '! Y -; 1 ' 43V m m ! ' ' ¢ 1

Y 1 - -; , ¤ < 7 5

) # ( $ 1?7A@ B mJ 5:' L 3 &-21 3 1A- ;- &1A--2 1 -; & m! ¢ ( ' 6. J m L 3 7 Y ¢ 3_f & -;&-21 3& 23 1A- $ a -;&-21 3P& 1A- - 0& ¤ &2- 1 1 m ! ( 7! ' 5n' L 7. 1 - -; A

J m ( ! ¤ Y -;& - a 1 & a

2 1 % G 3 L F F

2- 1 G J & 5.

:,93

+

m 'g5 ! 1 £ 43& -; A

( 7X! " Y $9% -;& - a 1 0& & J -21 £ £ Y 3 ¢ /

& l -

$

L1 N

L ¢ h 3 fI - P5 ! % Q( ! ]& - £ If

£ ¤ / - -

Y

m !(*' m 5 1 1

J L' £ Y 3 ¢ /

l -

10.

'

7 G L F J & £ & - -21 % P 5 ( ! ! #& - £ If

5n7 7 ( ! ( 1A- 3 !¢ & S1A L J ! S 3

11.

5_7 'g5 7! L 3 J Y 3P& - a2 S

7

Y

12.

m ! ( 7! ' ' % (*' L C 7 ( m ! ' 1 - A

J m " ( ! a $ ¤ Y

; & 2 1 G F3 J L - a -21 F G $9% &

8.

9.

1

£ ¤ / - -

43& 7 $ S - - a 1 3P&

' % ! J $ L 4 3P& a2S (] ! - S 43 & - 2- 1 0& 43& a2 1 :,!5

- 2- 1 0& ¤ -21

'I@CB = ?

5 '65 Y /

13.

1!

1

#$

( "' ' % (! ]% ! $

/ - a2S -21 3P&

5

!% $&% -; a 1 & & a2S ( ]! £ - S -21 43& % a2 1 P5 ! ?'I@CB Y -21 % Q( ! £ / - - = / - -21 7 7 G -;&-21 43& F SEa 1 (*' L ( & ' J

SE a 7 a 3 1 $ SE-;a &&-2- 1 3P& & -2- 1 a 1 & ¤ 1 " ( S 3 ¢ S £ 1A-; S -; J ' ! " L $ L J a 5n'65 ! '8>?7A@ B Y &-21 % $ $ 5n')( ! £ / - a 0 / = & 2 1

14.

1 ( !

'65 1

15.

5 1 ! 1 (* '

#$

#$

#$ 1 ( ! 16.

1 (*'

( &(' S '! % ( 7! ' % % $ 5n 9 $ % S -21 0 & X / 0 / a ( ! -21 5 ! £ £

; & 2 1 % a 3P& % ( ! / - &1

17.

1! 1 &

#$

£ & V 1 / A

- a ;- &-21 43& - -21 / 0 = _'I@ B

& ¤ F G , )

8.1.11. The hypergeometric function 1 F2 (a1 ; b1 , b2 ; z) 1.

7 ! 1N 5 1

(8 7 £ / N (: 7 ! 3 £ K3 ¢ 1 1

:, ;

( 1 £ / N

5 1

1

!

&

#$

¤

+

L1

( #$ 1 £ £ ! 1 N 5 ! 7 & / 1 N 5 ¤ 2. / 1 1

" " " C7( ! 3. Y 3 £ O - 9- £ 3 £ 3 £ / ?7 ' m ( 7 9BB B B !m " ( ! ! ! 7]&B B B $ 7 &BB B ' ! = @&#B 7! SEa a 1Rl a a 1 f f 3 ¢ - SEa a 1Rl 0 f 1Rl ) f , l 1 ) , >?7 ' m ( V( 7 V(9BB B B ! m " ( ! 7 &B B B ! $ 7 &B BB ' ! = @ #B O! ! f 0f 43 ¢ - SEa 1Rl 0f f S a a l a1

l 1) , , l a 1 ) >?7 ' m ( V( 7 V(9BB B B m " ( 7 &B B B ! $ 7 &B BB ' = @ #B ! !

3.

!

: 3 L

( 5 5 ' 6

0 f £f £ E S a R 1 l - l a 1 ) O 1 ¡\ £ , £f £ £ E S a - - a 1Rl

1 ¡\ £ , l 1 ) ( 5_7 P5 5 7 ! ( * ( ' (* m 5 ( ( 'g5 7 ! = @ 7X! B

0f

SEa R1 a l ¡ l £ -

4.

f 0f S Ea R1 a l ¢ £ !¡

l

£ - - SEa 1Rl £ O 1 ¡ £ f £ , l a 1 ) £ 3 £ - - SEl a a 1Rl 1 ) £ - O 1 ¡\ f £ O , ( 5_7 ! P5 5_7X! ( ( ' ! (* m 5 ( ( 'g5 7 ! = @ 7X! B

5.

6.

SEa a R1 l \¡ 0 f , l 1 ) £ E S a

l R1 a l ) 3 £ V-

SEl a 1Ra l )

£ £ ¡ 0 f £ f £ ¢ £ f £ 3 0 f ¡\ £ £ ¢ ( P 5_7 5 ! ( m 5 Q(

¢ £ £¢ £ ¢ £ £ 3 ¢ 5_7! (*'

,

£ , ( !

( ' * ! = @C7 O! B

8.2.2. Various Meijer G functions 1. 1 M 5 7! !¡

Y

K

7

7

*

_ 5 7

3

7 7 3 5_7 *

K

: 3 P

= M 5_7X! @CB

3.

4.

L1 , ,

1 ¡\ P3 f f 3 f f 7 £ £ £ £ ¤ / - / E3 / - /

A7 @CB 3 2 IF 7 G K 7 3 7 / ¢ 3h = 1& & ¡!¡ , * )

2.

3 / 0/ 0/ / 0/ 0 ¤ 1 1 1L ¢

¡!¡ J 3 2 l & 0 7 7 E(:7 7 7 E(:7 ( 7 O !C@CB Y K 3 3 K * = *

1 & ¢!¢!!¡

5.

6.

7.

8.

9.

10.

1 ¡\)¡\

3Pfg3

3Pfg3

f f f ) ¡ f

f f ! ¡

7 7

# 1 E 3

7

£ 1 ¤

f - 2- 1 £ / 0f a 1 f - -21 £ / f - -21

£ / ¤ £ / ¤

f 7 f

)¡ 1 - 0/ 3hf fT3 f £ £ 1 - / 1 - 0/ 0f

/ ¤

1 f f

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M]!

L @eV e J e w fK.e +e

' eV | j j fKs> ' M! eV | j j * j f 3j M! M! M! M! M! ; 5 M]!

"

M]! e+RJj L @ eV e L J B B B @eV}h| j @ e #e

+e

( KC7! J 7 C M! Z oe+K Z Jj' 3 #e.w 3j M ! Z " = 7R@ oe+w# 3j t 0 Z Jj* 3 # e.w 3j

s j ` 3 # e.w 3j

@eVK 3 #e.w 3j Z ( L oe+K@j 3 #e.w 3j

M]!

M! (

V

J

m $ ! Z ! @eVK f}>j .e`Jj j e # e.w 3j D ! :j JfR #e F +e

;M!

Z

Z

W @eV` +e f w

w fR #e

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&

:

#

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e Ve eS M]! L 5 7 ( ` @ eV |ZJj e e+ eS ! ^ J Z (* ! o e+ #w +e M! J

K

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M! oe+Rh| j j #w e+ f κ 5 M! & " κ 1 |eVS ee 3J | ! #eH# > w# fK.e

e "' ' ' M! ' M (87! ' eV B .w# ` j f 3j Ne+ .w# e eV M! M! B 1;- eS

:< ;

#

M! L J 1-

( M! M (87! ( ( M]! J L ! 99 L J 1;

9 ( ! ( 7! ( C7 ( ! ( ( ! 9 J 9 L

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M!

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5_7

565_7 " ! (65_7 9 ( L ! ( ! (

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Z "

M !

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M]! '%&

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M]!

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&

0eV%@Zs3 V j f 3j eV ' M! ' ! eS 1 1;- 'g 5?7X! ReV6@Zs3 V j Q ! ' M]! & 1;- f 3j eVK Z z

& " 1 M ] ! R + e n ee 3J } ! #e6# > w# fK.e κ κ5

e

" M]! = 'I@ ' M]! ' M! ! oe+K?@J f +e e+ |ZJj e e+`Z n ( @eVKs>.e +e ! =

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Z

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e = m !C@ "

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m " !

g $ m ! g $ & m !0! ( 5 ' 7

&B BB &B BB ' oe+R Z J o fKs j '

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w3f\f

#

M! ( @eVR Jf Ft .e e " M ! @eVRg A ett..e +e " ! 7 > @eVKg@ +e " m m ]]! " $ " "

"'& !! "5 " ]]! $ " " "

m "'& !! "5 ( ( )" " ' '( " ( 7X! J $ L $ e+Kue Vj w fKs> eVKZ z " M ! ! " "'& $ " " m #! !! " 5 $ "'&

" ]]! $ "" "'& !! "5 " #! " " "

!! " 5 "'& m ]! ! ! ! $ = " " 5 ! oe+ f ' ! #e# > w# fK.e e M]! K ! !( 9 M]! !( D !C@ K 9 =K 9 9 !;( M]! !( D !C@ K 9 =K ! 7 (* !( 7 ( 5 9 9 D K

]]! ! ! "5 m ]! ! ! "5 KM]! = M]!C@

m

"'& $ " " " "

" "'& " "

@eVK +e :

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