Differentially Transcendental Functions

DIFFERENTIALLY TRANSCENDENTAL FUNCTIONS

arXiv:math.GM/0412354 v1 17 Dec 2004

ˇ Zarko Mijajlovi´ c1) , Branko Maleˇ sevi´ c2) 1)

Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Belgrade, Serbia and Montenegro 2)

Faculty of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11000 Belgrade, Serbia and Montenegro Abstract The aim of this paper is to exhibit a method for proving that certain analytic functions are not solutions of algebraic differential equations. The method is based on model-theoretic properties of differential fields and properties of certain known transcendental differential functions, as of Γ(x). Furthermore, it also determines differential transcendency of solution of some functional equations.

1

Notation and preliminaries

The theory DF0 of differential fields of characteristic 0 is the theory of fields with additional two axioms that relate to the derivative D: D(x + y) = Dx + Dy,

D(xy) = xDy + yDx.

Thus, a model of DF0 is a differential field K = (K, +, ·, D, 0, 1) where (K, +, ·, 0, 1) is a field and D is a differential operator satisfying the above axioms. A. Robinson proved that DF0 has a model completion, and then defined DCF0 to be the model completion of DF0 . Subsequently L. Blum found simple axioms of DFC0 without refereing to differential polynomials in more than one variable, see [22]. In the following, if not otherwise stated, F, K, L, . . . will denote differential fields, F, L, K, . . . their domains while F∗ , K∗ , L∗ , . . . will denote their field parts, i.e. F∗ =(F, +, ·, 0, 1). Thus, F∗ [x1 , x2 , . . . , xn ] denotes the set of (ordinary) algebraic polynomials over F∗ in variables x1 , x2 , . . . , xn . The symbol L{X} denotes the ring of differential polynomials over L in the variable X. Hence, if f ∈ L{X} then for some n ∈ N , N = {0, 1, 2, . . .}, f = f (X, DX, . . . , Dn X) where f ∈ F∗ (x, y1 , y2 , . . . , yn ). Suppose L ⊆ K. The symbol td(K|L) denotes the transcendental degree of K∗ over L∗ . The basic properties of td are described in the following proposition. Proposition 1.1 Let A ⊆ B ⊂ C be ordinary algebraic fields. Then a. td(B|A) ≤ td(C|A). b. td(C|A) = td(C|B) + td(B|A). 1 Email

address : [email protected] address : [email protected] Second author supported in part by the project MNTRS, Grant No. 1861.

2 Email

1

The statement follows from the fact that every transcendental base of B over A can be extended to a transcendental base of C over A, see [16]. If b ∈ K, then L(b) denotes the simple differential extension of L in K, i.e. L(b) is the smallest differential subfield of K containing both L and b. Also, we shall use abbreviations: d.p. stands for differential polynomial. Thus, f is a d.p. over L in the variable X if and only if f ∈ L{X}. If L ⊆ K then b ∈ K is d.a. (differential algebraic) over L if and only if there is a non-zero d.p. f such that f (b, Db, D2 b, . . . , Dn b) = 0; if b is not d.a. over L then b is d.t. (differential transcendental) over L. K is a d.a. extension of L if every b ∈ K is d.a. over L, while d.e. stands for differential equation and a.d.e. stands for algebraic differential equation. Hence, f = 0 is a.d.e. if f ∈ L{X}. If f (X) ∈ F {X}\F , the order of f , denoted by ordf , is the largest n such that Dn X occurs in f . If f ∈ F we put ordf = −1. We shall write occasionally f ′ instead of Df . Models of DCF0 are differentially closed fields. A field K is differentially closed if and only if for every f ∈ L{X} if the equation f = 0 has a solution in some extension of K then it has a solution already in K. The theory DCF0 admits elimination of quantifiers and it is submodel complete (A.Robinson): if F ⊆ L, K then KF ≡ LF , i.e. K and L are elementary equivalent over F. Other notations, notions and results concerning differential fields that will be used in this article will be as in [22] or [19].

2

Differential algebraic extensions

In this section we shall state certain properties of extensions of differential fields. These properties parallel in most cases properties of algebraic fields (without differential operators), by replacing the notion of the algebraic degree with the notion of the transcendental degree of fields and degf by ordf . Proposition 2.1 Suppose L ⊆ K and let b ∈ K. Then b is d.a. over F if and only if td(L(b)|L) < ∞. Proof (⇒) Suppose b is d.a. over L. Then b satisfies a non-trivial a.d.e. f (y, Dy, D2 y, . . . , Dn y) = 0 where f ∈ L∗ (y, y1 , y2 , . . . , yn ). So b, Db, . . . , Dn b are algebraically dependent over L and td(L(b)|L) = ord m where m is a minimal polynomial of the differential ideal I(b|L) = {p ∈ K{X} : p(b) = 0} (see [19], Lemma 1.9., p.42), hence td(L(b)|L) < ∞. (⇐) Suppose td(L(b)|L) = n < ∞. Then b, Db, D2 b, . . . , Dn b are algebraically dependent over L, hence I(b, L) 6= {0}, i.e. b satisfies a nontrivial a.d.e. ∇ Suppose L ⊆ K and let a1 , a2 , . . . , an ∈ K be d.a. over L. Then L(a1 , a2 , . . . , ai ) = L(a1 , a2 , . . . , ai−1 )(ai ) 2

and ai is d.a. over L(a1 , a2 , . . . , ai−1 ) for each i = 1, 2, . . . , n. Thus, by Propositions 1.1 and 2.1 we have td(L(a1 , a2 , . . . , an )|L) =

td(L(a1 , a2 , . . . , an )|L(a1 , a2 , . . . , an−1 )) + . . . + td(L(a1 )|L) < ∞,

so every b ∈ L(a1 , . . . , an ) is d.a. over L. Hence L(a1 , . . . , an ) is d.a. over L. Proposition 2.2 Let L ⊆ K and suppose b, a1 , a2 , . . . , an ∈ K. If a1 , a2 , . . . , an are d.a. over L and b is d.a. over L(a1 , a2 , . . . , an ), then b is d.a. over L. Proof As L(b, a1 , a2 , . . . , an ) = L(a1 , a2 , . . . , an )(b), by Proposition 1.1 and the previous consideration, it follows that for some positive integers k, l td(L(b, a1 , a2 , . . . , an )|L) =

td(L(b, a1 , a2 , . . . , an )|L(a1 , a2 , . . . , an )) + td(L(a1 , a2 , . . . , an )|L) = k + l < ∞.

Therefore L(b, a1 , a2 , . . . , an ) is d.a. over L, hence b is d.a. over L either.

∇

Proposition 2.3 If F ⊆ L ⊆ K and L is d.a. over F and K is d.a. over L then K is d.a. over L. Proof Suppose that b ∈ K. As b is d.a. over L there is a d.p. f ∈ L{X} such that f ∈ I(b|L). Let a1 , a2 , . . . , an ∈ L be all constants from L appearing in f . Then, obviously, b is d.a. over F (a1 , a2 , . . . , an ), hence b is. d.a. over F. ∇ Proposition 2.4 Suppose L ⊆ K and b ∈ K. Then b is a d.a. over L if and only if Db is d.a. over L. Proof (⇒) Suppose b ∈ K is d.a. over L. Then there is a non-trivial f ∈ L{X} such that f (b, Db, . . . , Dn b) = 0 for some natural n. Then g(Db, . . . , Dn−1 (Db)) = 0

where g(y1 , . . . , yn ) ∈ L(b){X}

and g(y1 , . . . , yn ) = f (b, y1 , . . . , yn ). Hence, Db is d.a. over L(b), so Db is d.a. over L as b is d.a. over L. (⇐) If Db is d.a. over L, then there is a a.d.e. f = 0 over L satisfied by b, hence b is d.a. over L. ∇ Theorem 2.5 Suppose F ⊆ K and let L = {b ∈ K : b is d.a. over F}. Then a. L is a differential subfield of K extending d.a. F. b. If K is d.a. closed then L is d.a. closed. Proof a. Suppose that a, b ∈ L. As a, b are d.a. over F, it follows that F(a, b) is d.a. extension of F as well. But a + b, ab ∈ F (a, b) and b−1 ∈ F (a, b) if b 6= 0, hence L is closed under field operations, i.e. L is a subfield of K extending d.a. F. L is closed under differentiation D by Proposition 2.4. 3

b. Let f ∈ L{X} and suppose that f = 0 has a solution in an extension of L. Then f = 0 has a solution in a differentially closed field H ⊇ L. As HL ≡ KL , i.e. H and K are elementary equivalent over L, it follows that f = 0 has a solution in K. Therefore, there are a1 , a2 , . . . , an ∈ L, and b ∈ K such that f ∈ F (a1 , a2 , . . . , an ) and f (b) = 0. So b is d.a. over F (a1 , a2 , . . . , an ) and a1 , a2 , . . . , an are d.a. over F, so by Proposition 2.2 b is d.a. over F. Hence b belongs to L, i.e. f = 0 has a solution in L, thus L is differentially closed. ∇ Let us denote by MD the class of complex functions meromorphic on a complex domain D. If D = C then we shall write M instead of MD . Let C = C(z) be the differential field of complex rational functions. Then M is differential field and C ⊆ M. Further, let L = {f ∈ M : f d.a. over C}. By Theorem 2.5 L is a differential subfield of M extending d.a. C. Assume that e, g are complex functions. If e is an entire function (i.e. holomorphic in entire finite complex plane) and g is meromorphic, then their composition e ◦ g is not necessary meromorphic, as the example e(z) = ez , g(z) = 1/z shows. However, g ◦ e is meromorphic since g is a quotient of entire functions, and obviously entire functions are closed under compositions. The next proposition states that a similar property holds for a.d. functions. Proposition 2.6 Let e, g be complex a.d. functions over C. If e is entire and g is meromorphic, then g ◦ e is meromorphic and a.d. over C. Proof Suppose that g is a solution of an a.d.e. over C, i.e. f (z, g, Dg, . . . , Dn g) ≡ 0 for some f ∈ C[‡, ⊓′ , ⊓∞ , . . . , ⊓\ ]. ′ If e = 0 then e is a constant, so we may assume the nontrivial case, that e′ 6= 0. There are rational expressions λkj in e′ , e′′ , . . . such that (Dk u) ◦ e =

k X

λkj Dj (u ◦ e),

u is any meromorphic function.

j=1

In fact, the sequence λkj satisfies the recurrent identity λk+1,j = (λ′kj + λk,j−1 )/e′ , λ11 = 1/e′ , λk0 = 0, λkj = 0 if j > k. It is easy to see that λkk = (1/e′ )k . Let f1 = f (e, y, λ11 Dy, λ21 Dy + λ22 D2 y, . . . , λn1 Dy + . . . + λnn Dn y). Pk Then g ◦ e is a solution of f1 = 0: since (Dk g)◦ e = j=1 λkj Dj (g ◦ e), it follows, taking h = g ◦ e, f1 (z, h, Dh, . . . , Dn h)

= f (e, g ◦ e, (Dg) ◦ e, . . . , (Dn g) ◦ e) = f (z, g, Dg, . . . , Dn g)|z=e ≡ 0.

Therefore, g ◦ e is meromorphic and d.a. over C. ∇ The next theorem is a direct corollary of the previous proposition and Proposition 2.4, since L is a field and therefore it is closed under values of rational expressions. 4

Theorem 2.7 Let a(z) be a complex differential transcendental function over C, f (z, u0 , u1 , . . . , um , y1 , . . . , yn ) a rational expression over L, and assume that e1 , e2 , . . . , en are entire functions which are d.a. over C, e′i 6= 0, 1 ≤ i ≤ m. If b is meromorphic and f (z, b, b ◦ e1 , . . . , b ◦ em , Db, . . . , Dn b) ≡ a(z) then b is d.t. over C. Let us notice that N. Steinmetz in the paper [24] (Satz 5.) has considered conversely theorem of the previous. We shall use it in proofs of differential transcendentality of certain complex functions. In most cases functions ei (z) will be linear functions αz + β, α 6= 0, and f will be a polynomial over C (observe that C ⊂ L). Also, it is possible to consider differential transcendental functions when f is a rational expression ′′ ′ over L, for example as a solution of the equation y (z) + y (z)/y(z) + y(sin z) = Γ(z +1).

3

Differentially transcendental functions

Suppose L ⊆ K. Let R(x) be the differential field of real rational functions. The following H¨ older’s famous theorem asserts the differential transcendentality of Gamma function, see [10]. Theorem 3.1 a. Γ(x) is not d.a. over R(x). b. Γ(z) is not d.a. over C(z). ∇ Now we shall use the transcendentality of Γ(z) and properties of differential fields developed in the previous section to prove differential transcendentality over C of some analytic functions. Γ(z) is meromorphic and by H¨ older’s theorem and Proposition 2.3, Γ(z) 6∈ L. Example 3.2 This example is an archetype of proofs, based on properties of differential fields exhibited in the previous section, of differential transcendentality of some well known complex functions. The Riemann zeta function is differentially transcendental over C (Hilbert). First we observe that ζ(s) is meromorphic and that ζ(s) satisfies the well-known functional equation: ζ(s) = χ(s)ζ(1 − s),

where χ(s) = 

(2π)s 2Γ(s) cos(

2 .

πs ) 2

Now, suppose that ζ(s) is d.a. over C, i.e. that ζ(s) ∈ L. Then ζ(1 − s) and ζ(s)/ζ(1 − s) belong to L, too, so χ(s) belongs to L. The elementary functions 2 (2π)s , and (cos( πs 2 )) obviously are d.a. over C i.e. they belong to L. As L is a field, it follows that Γ(z)2 belong to L, so Γ(z) is d.a. over L. Hence, Γ(z) ∈ L but this yields a contradiction. Therefore ζ(s) is differentially transcendental

5

function over C. Generally, Dirichlet L-series Lk (s) =

∞ X

χk (n)

n=1

1 ns

(k ∈ Z),

where χk (n) is Dirichlet character see [13], is differentially transcendental function over C. This follows from a well-known functional equations L−k (s) = 2s π s−1 k −s+1 Γ(1 − s) cos( and L+k (s) = 2s π s−1 k −s+1 Γ(1 − s) sin(

πs )L−k (1 − s) 2

πs )L+k (1 − s). 2

Besides Riemann zeta function ζ(s) = L+1 (s), Dirichlet eta function η(s) =

∞ X

(−1)n+1

n=1

1 = (1 − 21−s )L+1 (s) ns

and Dirichlet beta function β(s) =

∞ X

(−1)n

n=0

1 = L−4 (s) (2n + 1)s

are transcendental differential functions as examples of Dirichlet series see [2] (p.289). ∇ Example 3.3 Kurepa’s Function related to the Kurepa Left Factorial Hypothesis, see [6] (problem B44), is defined by (1)

K(z) =

Z∞

e−t

tz − 1 dt. t−1

0

Kurepa established in [15] that K(z) can be continued meromorphically to the whole complex plane. Also, it satisfies the recurrence relation (2)

K(z) − K(z − 1) = Γ(z),

hence, by Theorem 2.7, K(z) is d.t. over C. The functional equation (2), besides Kurepa’s function K(z), has another solution which is given by the series (3)

K1 (z) =

∞ X

Γ(z − n).

n=0

This function has simple poles at integer points. We can conclude similarly that K1 (z) is transcendental differential function. Between functions K(z) and K1 (z) the following relation is true see [23] and [17]: K(z) =

Ei(1) π − ctg πz + K1 (z), e e 6

where Ei(x) is the exponential integral see [7]. Each meromorphic solution of the functional equation (2) is d.t. over C. Alternating Kurepa’s function related to the alternating sums of factorials see [6] (problem B43), is defined by (4)

A(z) =

Z∞

e−t

tz+1 − (−1)z t dt. t+1

0

A(z) is meromorphic and it satisfies the functional equation (5)

A(z) + A(z − 1) = Γ(z + 1).

The functional equation (5), besides alternating Kurepa’s function A(z), has another solution which is given by the series (6)

A1 (z) =

∞ X

(−1)n Γ(z + 1 − n).

n=0

As in the case of K(z), we see that A(z) and A1 (z) are d.t. over C. The following identity holds, see [18]:
πe + A1 (z). A(z) = − 1 + e Ei(−1) (−1)z + sin πz Generally, each meromorphic solution of a functional equation (5) is transcendental differential function over the field C. G.V.Milovanovi´c introduced in [20] a sequence of meromorphic functions Km (z) − Km (z − 1) = Km−1 (z),

K−1 (z) = Γ(z), K0 (z) = K(z).

By use of induction and Theorem 2.7 we can conclude that Km (z) are d.t. over C. Analogously, a sequence of meromorphic functions defined by Am (z) + Am (z − 1) = Am−1 (z),

A−1 (z) = Γ(z + 1), A0 (z) = A(z)

is a sequences of transcendental differential functions over C.

∇

Example 3.4 The meromorphic function H1 (z) =

∞ X

1 (n + z)2 n=0

is differentially transcendental over C. Really, D2 ln(Γ(z)) = H1 (z), i.e. Γ(z) satisfies the a.d.e. (D2 Γ)Γ − (DΓ)2 − H1 Γ2 = 0 over C(H∞ ). Thus, if H1 would be d.a. over C, then by Proposition 2.3, Γ would be too, what yields a contradiction. Hence, H1 (z) is differentially transcendental function over C. ∇ Remark 3.5 We see that in Example 3.4 functions Γ(z) and H1 (z) are diffe7

rentially algebraically dependent, i.e. g(Γ, H1 ) = 0, where g(x, y) = x′′ x − (x′ )2 − yx2 . We do not know if similar dependencies exist for pairs (K, Γ) and (ζ, Γ). It is very likely that these pairs are in fact differentially transcendental. Example 3.6 W.Gautschi and J.Waldvogel considered in [5] a family of functions which satisfies Iα (x) = xIα−1 (x) + Γ(α),

α > −1, x > 0.

For each x0 > 0 function f (α) = Iα (x0 ) can be continued meromorphically (with respect to the complex α, see [5]). By Theorem 2.7 the function f (α) is d.t. over C. ∇ Example 3.7 The Barnes G-function is defined by [1]:  ∞  Y 2 z n −z+z2 /(2n) , e G(z + 1) = (2π)z/2 e−(z(z+1)+γz )/2 1+ n n=1 where γ is the Euler constant. Barnes G-function is an entire function and it satisfies following functional equation (7)

G(z + 1) = Γ(z)G(z),

By Theorem 2.7 each meromorphic solution of the functional equation (7) is d.t. over C. ∇ Example 3.8 The Hadamard’s factorial function is defined by:    1 z d 1 − z  H(z) = Γ / Γ 1− , Γ(z − 1) dz 2 2 see [3]. Hadamard’s function is an entire function and it satisfies following functional equation (8)

H(z + 1) = zH(z) +

1 . Γ(1 − z)

Using the method from Example 3.3 we can conclude that each meromorphic solution of a functional equation (8) is transcendental differential function over the field C. ∇ ∞ ∞ X Y Example 3.9 Let τ (n) be defined by τ (n)q n = q (1 − q n )24 , q = e2πiz . n=1

n=1

This function was first studied by [21]. Ramanujan’s Dirichlet L-series defined by ∞ X τ (n) , f (s) = ns n=1

is differentially transcendental function over C. This follows from functional

8

equation see [8] (p.173): f (s) =

i.e. f (s) = −

(2π)s (2π)12−s

(2π)s Γ(12 − s) f (12 − s) (2π)12−s Γ(s)  Y 11 1 π (s − i) f (12 − s). sin(πs) i=1 Γ(s)2

∇

Example 3.10 Let us note that function Z∞ p−1 x dx = Γ(p)ζ(p) ex − 1 0

is differentially transcendental over
C. This follows from the functional equation ζ(1−s) = 2(2π)−s cos( sπ 2 ) Γ(s)ζ(s) . Analogously function Γ(p)β(p) is differentially transcendental over C. In this example we give some integrals from tables of integrals [4] and [7] which determine differentially transcendental functions over C. f (p) Z∞ 0

Z∞

x Γ(p) dx = p ζ(p) eax − 1 a p−1

p−1

x dx = sh(ax)

0

Z∞

Γ(p) 

x dx = p eax + 1 a

0

Z∞

parameters

p−1

2 1 − p ζ(p) 2

2Γ(p)  ap

(Re p > 0, Re a > 1)

1 1 − p ζ(p) 2

(Re p > 0 , Re a > 0)

(Re p > 1 , a > 0)

p−1

2Γ(p) x dx = β(p) ch(ax) ap

(Re p > 0 , a > 0)

0

Z∞

xp−1 e−ax sin(mx) dx =

Γ(p) sin(p θ) (a2 + m2 )p/2

xp−1 e−ax cos(mx) dx =

Γ(p) cos(p θ) (a2 + m2 )p/2

0

Z∞ 0

Z∞

sin θ = m/r, cos θ = a/r, r = (a2 + m2 )1/2 , (Re p > −1 , Re a > |Im m|) sin θ = m/r, cos θ = a/r, r = (a2 + m2 )1/2 , (Re p > 0 , Re a > |Im m|)

references [4] 860.39 [7] 3.411/1 [4] 860.49 [7] 3.411/3 [4] 860.509 [7] 3.523/1 [4] 860.539 [7] 3.523/3 [4] 860.89 [7] 3.944/5 [4] 860.99 [7] 3.944/6

xp−1 e−ax sh(mx) dx

0

(Re p > −1 , Re a > |Re m|)

[7] 3.551/1

0

(Re p > 0 , Re a > |Re m|)

[7] 3.551/2

  1 = Γ(p) (a − m)−p − (a + m)−p 2 Z∞ xp−1 e−ax ch(mx) dx   1 = Γ(p) (a − m)−p + (a + m)−p 2

Let us remark that using this method we can conclude which integrals from other tables of integrals are differentially transcendental functions over C. ∇ 9

References [1] Barnes,E.W., The Theory of the G-Function, Quart. J. Pure Appl. Math. 31, (1900) 264–314. [2] Borwein,J.M., Borwein,P.B., Pi & the AGM : A Study in Analytic Number Theory and Computational Complexity, (1987) John Wiley & Sons, New York. [3] Davis,P.J., Leonhard Euler’s Integral: a historical profile of the gamma function, Amer. Math. Monthly 66, (1959) 849–869. [4] Dwight,H.B., Tables of Integrals and Other Mathematical Data, 4th ed., (1961) The Macmillan Co., New York. [5] Gautschi,W., Waldvogel,J., Computing the Hilbert transform of the generalized Laguerre and Hermite weight functions, BIT 41:3, (2001) 490–503. [6] Guy,R.K., Unsolved problems in number theory, 2nd ed., (1994) SpringerVerlag. [7] Gradxten,I.S., Ryжik,I.M. Tablicy integralov, summ, rdov i proizvedeni, izdanie ptoe, (1971) Nauka, Moskva. [8] Hardy,G.H., Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed., (1959) Chelsea, New York. [9] Hurwitz,A., Courant,R., Vorlesungen ueber allgemeine Funktionentheorie und elliptische Funktionen, 4th ed., (1964) Springer Verlag . ¨ [10] H¨ older,O., Uber die Eigenschaft der Gamma Funktion keiner algebraische Differentialgleichung zu gen¨ ung, Math. Ann., 28, (1887) 1-13. [11] Ivi´c,A., The Riemann zeta-function, (1985) John Wiley & Sons, New York. ˇ On Kurepa problems in number theory, Publ. Inst. [12] Ivi´c,A., Mijajlovi´c,Z., Math. (N.S.) 57 (71), (1995) 19–28. [13] Ireland,K., Rosen,M., A Classical Introduction to Modern Number Theory, (1982) Springer-Verlag. [14] Kolchin,E.R., Differential Algebra and Algebraic Groups, (1973) New York, Academic Press. - ., Left factorial in complex domain, Mathematica Balkanica 3, [15] Kurepa,D (1973) 297–307. [16] Lang,S., Algebra, 3rd ed., (2002) Springer-Verlag. [17] Maleˇsevi´c,B., Some considerations in connection with Kurepa’s function, Univ. Beograd. Publ. Elektrotehn. Fak., Ser. Mat., 14, (2003) 26–36. 10

[18] Maleˇsevi´c,B., Some considerations in connection with alternating Kurepa’s function, preprint, (2004) http://arXiv.org/abs/math/0406236. [19] Marker,D., Messmer,M., Pillay,A., Model Theory of Fields, (1996) SpringerVerlag. [20] Milovanovi´c,G., A sequence of Kurepa’s functions, Scientific Review, Ser. Sci. Eng. 19-20, (1996) 137–146. [21] Ramanujan,S., On certain arithmetical functions, Trans. Cambr. Phil. Soc. 22, (1916) 159–184. [22] Sacks,G.E., Saturated Model Theory, (1972) W. A. Benjamin, Reading, Massachusetts. [23] Slavi´c,D., On the left factorial function of the complex argument, Mathematica Balkanica 3, (1973) 472–477. ¨ [24] Steinmetz,N., Uber die faktorisierbaren L¨ osungen gew¨ ohnlicher, Mathematische Zeitschrift, 170, (1980) 169–180.

11