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Stirling functions of first kind in the setting of fractional calculus and generalized differences P. L. Butzer, A. A. Kilbas

b ¶

Email: [email protected] , L. Rodríguez-Germá

[email protected] & J. J. Trujillo a

c #

c !

Email:

Email: [email protected]

Lehrstuhl A für Mathematik, RWTH Aachen, Templergraben 55, D-52056, Aachen, Germany

b

Department of Mathematics and Mechanics, Belarusian State University, 220050, Minsk, Belarus c

Departamento de Análisis Matemático, Universidad de La Laguna, 38271, La LagunaTenerife, Spain Available online: 21 Jun 2011

¶

!

To cite this article: P. L. Butzer, A. A. Kilbas Email: [email protected] , L. Rodríguez-Germá Email: [email protected] & J. J. #

Trujillo Email: [email protected] (2007): Stirling functions of first kind in the setting of fractional calculus and generalized differences, Journal of Difference Equations and Applications, 13:8-9, 683-721 To link to this article: http://dx.doi.org/10.1080/10236190701470225

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Journal of Difference Equations and Applications, Vol. 13, Nos. 8–9, August –September 2007, 683–721

Stirling functions of first kind in the setting of fractional calculus and generalized differences

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´ §k and J. J. TRUJILLO§# P. L. BUTZER†*, A. A. KILBAS‡{, L. RODRI´GUEZ-GERMA †Lehrstuhl A fu¨r Mathematik, RWTH Aachen, Templergraben 55, D-52056 Aachen, Germany ‡Department of Mathematics and Mechanics, Belarusian State University, 220050 Minsk, Belarus §Departamento de Ana´lisis Matema´tico, Universidad de La Laguna, 38271 La Laguna-Tenerife, Spain (Received 9 July 2006; revised 4 November 2006; in final form 5 March 2007) Dedicated to the Memory of Professor Bernd Aulbach**

The purpose of this paper is to present a new approach to generalizations of Stirling numbers of the first kind by the application of differential and integration operators of fractional order and generalized, infinite differences. Such an approach allows us to extend the classical Stirling numbers of the first kind, s(n, k), to functions s(a, b), where both parameters n, k have been extended to complex a, b. Under such a construction the s(a, b) turn out to have the series representation—a major result of this paper

sða; bÞ ¼

1 ebpi X ð21Þ j Gð2aÞ j¼0

2a 2 1 j

!

1 ð j þ 1Þbþ1

for Re b . Re a, with sða; 0Þ ¼ 1=Gð1 2 aÞ for any a [ C when b ¼ 0. Various properties of the new Stirling functions are established, most generalize those known for the numbers s(n, k); some are new, i.e. a multiple sum formula for s(a, k), and an interesting connection between the s(a, b) and the Riemann zeta function zðb þ 1Þ for complex b with Re b . 0. Several connections between the s(a, b) and the Stirling functions of second kind, s(a, b), studied earlier by the authors, are deduced. Thus the s(2n, b) coincide with the Stirling functions S(2 b, n) of second kind, apart from a multiplicative constant. Of fundamental importance is the orthogonality property of the s(a, k) and S(k, m). The basic tool here is the Shannon sampling theorem of signal analysis. The Riemann– Liouville fractional derivative is expressed in terms of Hadamard derivatives, which involve the powers of the operator d ¼ x(d/dx). The sampling representation of the Mittag– Leffler function E1;12a ðlxÞ=Gða þ 1Þ as a function of a is one of the many new results. Finally, a new “infinite” or fractional order difference operator, Da, is defined in terms of the s(a, k); it involves the powers of the operator Q ¼ xD. This calculus of “infinite” differences is applied to representative examples, including the factorial and exponential functions. Keywords: Stirling numbers and Stirling functions of first and second kind; Riemann zeta function; Differences of fractional order; Riemann–Liouville and Hadamard fractional derivatives; Shannon sampling theorem; Factorial and exponential functions Mathematics Subject Classification: 33E99; 11B73; 05A10; 11M06; 26A33; 94A20

*Corresponding author. Email: [email protected] {Email: [email protected] kEmail: [email protected] #Email: [email protected] **This paper is dedicated to Bernd Aulbach in recognition of his many, basic contributions to mathematics, to differential and difference equations, to chaos theory, to dynamical systems. The senior author (Butzer) met Bernd only a few months before his unexpected death at a personal meeting with Saber Elaydi and Ulrich Eckern in Augsburg. Journal of Difference Equations and Applications ISSN 1023-6198 print/ISSN 1563-5120 online q 2007 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/10236190701470225

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1. Introduction

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The classical Stirling numbers s(n, k) of the first kind, introduced by James Stirling in his famous work Methodus Differentialis of 1730 [1], play, together with the Stirling numbers of second kind, the S(n, k), an important role in the calculus of finite differences, in combinatorial problems, in numerical analysis, interpolation theory and number theory. Those of first kind, the s(n, k), can be defined in terms of their (horizontal) generating function ½z "n ¼

n X

sðn; kÞz k

k¼0

ðz [ C; n [ N0 Þ;

ð1Þ

where ½z "n U zðz 2 1Þ . . . ðz 2 n þ 1Þ is the falling factorial polynomial, thus equivalently by ! " 1 d k# sðn; kÞ ¼ x "n jx¼0 k! dx

ðk [ N0 Þ:

A further equivalent approach is via their exponential generating function 1 ½logð1 þ zÞ"k X zn ¼ sðn; kÞ k! n! n¼k

ðjzj , 1; k [ N0 Þ;

thus, in view of the Taylor expansion, also by ! " $ 1 d n# logð1 þ xÞ"k $x¼0 sðn; kÞ ¼ k! dx

ðx [ RÞ:

ð2Þ

ð3Þ

The Stirling functions of the first kind, the s(a, k), where n [ N is extended to real a [ R as well as to complex a [ C, first studied from 1989 on in [2– 5], can be defined in terms of the infinite sum ½z "a ¼

1 X k¼0

sða; kÞz k

ðjzj , 1; a [ CÞ;

ð4Þ

since [z ]a is holomorphic for jzj , 1, where ½z "a U Gðz þ 1Þ=Gðz þ 1 2 aÞ (a [ CnZ2 ), is the falling factorial function, thus also equivalently by sða; kÞ ¼

! " $ 1 d k $ ½x "a $ k! dx x¼0

ða [ C; k [ N0 Þ:

ð5Þ

The Stirling functions of first kind, s(a, k), were also defined by the fractional counterpart of (3), namely sða; kÞ ¼

% & 1 lim Da0þ ½logðtÞ"k ðxÞ k! x!1

ðReðaÞ . 0; k [ N0 Þ:

ð6Þ

Here Da0 þ is the Riemann – Liouville fractional differentiation operator of order a [ C (ReðaÞ $ 0; a – 0), defined for n ¼ ½ReðaÞ" þ 1 by %

& Da0þ f ðxÞ U

! "n d % n2a & I 0þ f ðxÞ ðx . 0; ReðaÞ $ 0; a – 0Þ; dx

D00þ f ðxÞ ¼ f ðxÞ;

ð7Þ

Stirling functions of the first kind. Fractional Calculus

685

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where I a0 þ is the Riemann – Liouville fractional integration operator of order a [ C (Re(a) . 0), thus ð ! a " 1 x f ðtÞdt ðx . 0Þ; ð8Þ I 0þ f ðxÞ U GðaÞ 0 ðx 2 tÞ12a (see Samko et al. [16], Sections 2.3 and 2.4). In particular, when a ¼ n, definition (6) coincides with that of (3) since $ %n ! n " d k D0þ ½logðtÞ& ðxÞ ¼ ½logðxÞ&k ðn [ N0 Þ: dx

With this approach we can also define the s(a, k) for ReðaÞ , 0 by sða; kÞ U

! " 1 lim I 2a ½logðtÞ&k ðxÞ k! x!1 0þ

ðk [ ZÞ:

ð9Þ

The first important result of this paper, namely Theorem 2.1, is that the Stirling functions s(a, k), defined by (6) and (9), coincide with those given by definition (5). Thus these definitions are equivalent. Returning to the work of Butzer et al. [5], two of their basic results are the following: Theorem 1.1 (Representation theorem). For a [ C and k . Re(a) (k [ N), one has ð1 1 1 ½log u &k sinðapÞ X Gða þ jÞ du ¼ ð21Þkþ1 : ð10Þ sða; kÞ ¼ a þ1 Gð2aÞk! 0þ ð1 2 uÞ ð j 2 1Þ! j kþ1 p j¼1 For a proof, which makes use of the sampling theorem for s(a, k), thus a representation of the function sða; kÞ=Gða þ 1Þ for a [ C in terms of the numbers sð j; kÞ=j! for j [ N0; ([5], p. 13– 15, and [17], p. 100, and section 5.2 below). The second result to follow, which is also important for the present paper, will be included with a proof (see also p.15 of [5]). It is based on definition (4) of s(a, k) and properties of the polygamma function ([24], Section 1.16). Theorem 1.2 (Recursion formula). sða; k þ 1Þ ¼

If a [ C and k [ N, then

k 1 X c ðk2jÞ ð1Þ 2 c ðk2jÞ ð1 2 aÞ sða; jÞ ða ! NÞ; k þ 1 j¼0 ðk 2 jÞ!

where c ðmÞ is the m-th polygamma function, i.e., $ %m d c ðmÞ ðzÞ ¼ c ðzÞ dz

ðz [ CnZ0 Þ;

ð11Þ

ð12Þ

and cðzÞ U G0 ðzÞ=GðzÞ is the digamma function. Proof. The function Fðx; aÞ U cðx þ 1Þ 2 cðx þ 1 2 aÞ for any x [ D U Cn{x [ R; x 2 a [ Z2 } is holomorphic in D (see p.261 of [20]), thus in jxj , e , for e small, a ! N. Hence

686

P. L. Butzer et al.

it can be expanded as a power series about x0 ¼ 0 : Fðx; aÞ ¼

1 X k¼0

Ck ðaÞx k

for jxj , e ; a ! N, where Ck ðaÞ U

c ðkÞ ð1Þ 2 c ðkÞ ð1 2 aÞ k!

ðk [ N0 Þ;

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noting that the function Ck(a) equals C k ð aÞ ¼

! " # 1 d k # Fðx; aÞ# : k! dx x¼0

Further, differentiating the series (4) for jxj , 1 yields

1 X d ½x %a ¼ ðk þ 1Þsða; k þ 1Þx k : dx k¼0

ð13Þ

But by definition the left-hand derivative equals $ % d Gðx þ aÞ G0 ðx þ 1ÞGðx þ 1 2 aÞ 2 Gðx þ 1ÞG0 ðx þ 1 2 aÞ ¼ dx Gðx þ 1 2 aÞ ½Gðx þ 1 2 aÞ%2 & ' & ' 0 G0 ðx þ 1Þ Gðx þ 1Þ Gðx þ 1Þ G ðx þ 1 2 aÞ ; ð14Þ ¼ 2 Gðx þ 1Þ Gðx þ 1 2 aÞ Gðx þ 1 2 aÞ Gðx þ 1 2 aÞ thus, by power series multiplication, for jxj , 1, ! $ % 1 k X X d Gðx þ aÞ sða; jÞCk2j ðaÞ x k : ¼ ½x %a Fðx; aÞ ¼ dx Gðx þ 1 2 aÞ j¼0 k¼0

ð15Þ

Comparing coefficients of the series (13) and (15) yields the theorem. In what follows we need to recall the definition of the power function x b, defined for real x [ R; x – 0, and b [ C, namely x b ¼ exp{b½logjxj þ i arg x %}ð2p % arg x , pÞ

ð16Þ

ebpi ¼ e2pImðbÞ {cosðp ReðbÞÞ þ i sinðp ReðbÞÞ}:

ð17Þ

The actual purpose of this paper is to generalise the Stirling functions s(a, k) with a [ C; k [ N0 to functions s(a, b) where both a and b are complex. In two previous papers [10,15] the authors have already studied the extension of the Stirling functions of second kind, the S(a, k), to the functions S(a, b), where both a, b were complex. Thus for a [ C (a – 0), k [ N or b [ C (b – n þ 1; n þ 2; · · ·), with ReðbÞ . n and n [ N0 , we had, respectively, ! ! k 1 k b X 1X 1 k2j a j Sða; kÞ ¼ ð21Þ ð21Þ ðb 2 jÞn : ð18Þ j ; Sðn; bÞ ¼ j j k! j¼1 Gðb þ 1Þ j¼0 As to the s(a, b), we have, in generalisation of (6),

A

Stirling functions of the first kind. Fractional Calculus

Definition 1.3.

Let a and b belong to C. Then

sða; bÞ U

! " 1 lim Da0þ ½logðtÞ%b ðxÞ ðReðaÞ $ 0; a – 0Þ; Gðb þ 1Þ x!1

sð0; bÞ U sða; bÞ U

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687

! " 1 lim D00þ ½logðtÞ%b ðxÞ Gðb þ 1Þ x!1 ! a " 1 b lim I 2 0þ ½logðtÞ% ðxÞ Gðb þ 1Þ x!1

ð19Þ

ðReðbÞ . 0Þ;

ð20Þ

ðReðaÞ , 0Þ:

ð21Þ

The chief result of this paper is Theorem 3.4 of Section 3. In order to understand the background of our approach it may be advisable to phrase this theorem at this stage somewhat differently, in two parts. Theorem 1.4. (a) Let a; b [ C such that ReðaÞ , 0, ReðbÞ % 0. Then ð 12 1 ð1 2 tÞ2a21 ½logðtÞ%b dt sða; bÞ ¼ Gðb þ 1ÞGð2aÞ 0þ ! 1 2a 2 1 ebpi X 1 j ð21Þ : ¼ j Gð2aÞ j¼0 ð j þ 1Þbþ1

ð22Þ

(b) Let a, b [ C such that Re(a) $ 0 and ReðbÞ . ReðaÞ, with n ¼ ½ReðaÞ% þ 1. Then $ %n ðx 1 › 1 lim sða; bÞ ¼ ðx 2 tÞn2a ½logðtÞ%b dt Gðb þ 1Þ x!1 ›x Gðn 2 aÞ 0 ! 1 2a 2 1 ebpi X 1 j ¼ ð21Þ : ð23Þ j Gð2aÞ j¼0 ð j þ 1Þbþ1 $ % a Above, b are, of course, the binomial coefficients defined for complex a, b [ C, a – 2 1, 2 2, . . . , and for integers n, j [ N0, respectively, by ! ! n a Gða þ 1Þ n! ; ¼ ¼ j Gðb þ 1ÞGða 2 b þ 1Þ j!ðn 2 jÞ! b

ð j # nÞ:

ð24Þ

Remark 1. Theorem 1.4 yields the same series representation in (22) and (23) for s(a, b) with complex a, b [ C provided that ReðbÞ . ReðaÞ. Remark 2. The series in (10) coincides with the series in (23) (thus (70) below) in case b ; k . ReðaÞ, noting the relations (see, Erdelyi et al. [24], 1.2(6) and 1.2(2)): GðzÞGð1 2 zÞ ¼

p ; sinðp zÞ

ðzÞk ¼

Gðz þ kÞ ; GðzÞ

688

P. L. Butzer et al.

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with z [ C and k [ N. Thus (23) is a generalization of (10) to the case that k [ N is replaced by b [ C (ReðbÞ . ReðaÞ). The paper is organized as follows. Section 2 deals with properties of s(a, k) for k [ N0 expressed in terms of digamma functions, as multiple sum representations or as sums involving the s( j, k) for j [ N. Section 3 is devoted to our new infinite series representation for the general Stirling functions s(a, b), thus the proof of Theorem 1.4. The differentiability of s(a, b) as functions of a and b in C is also treated here, as well as new connections of the s(a, b) with the Riemann function z(b þ 1) when a ! 0. Section 4 concerns several recurrence relations. Section 5 is devoted to connections between the Stirling function of first kind and those of second kind, studied in two earlier papers [10 and 15]. Fundamental here is the orthogonality relation between the s(a, k) and S(k, m), the proof of which depends upon the Shannon sampling theorem of signal analysis. As further applications Section 6 concerns connections between the Riemann – Liouville and Hadamard fractional derivatives and integrals. Finally, a fractional difference operator Da is studied, connecting it with the operator u ¼ xD and its powers, in terms of the s(a, k). Five representative examples are presented, including the factorial and exponential functions. 2. Properties of the functions s(a, k) In this section we will present some important properties of the Stirling functions s(a, k). 2.1 Expressed in terms of falling factorial functions We first show that the Stirling functions s(a, k), a [ C, defined by (6) and (9), coincide with those given by (5), thus that the two definitions are equivalent. Theorem 2.1.

Let a [ C and k [ N0.

(a) If ReðaÞ $ 0, then

# $k % & ! " 1 1 d Gðx þ 1Þ lim Da0þ ½logðtÞ&k ðxÞ ¼ lim Gðk þ 1Þ x!1 k! x!0 dx Gðx þ 1 2 aÞ # $k 1 d ½x &a : ¼ lim k! x!0 dx

sða; kÞ ¼

ð25Þ

(b) If ReðaÞ , 0, then sða; kÞ ¼

# $k % & ! a " 1 1 d Gðx þ 1Þ k lim I 2 lim ðxÞ ¼ ½logðtÞ& : Gðk þ 1Þ x!1 0þ k! x!0 dx Gðx þ 1 2 aÞ

ð26Þ

In particular, sða; 0Þ ¼

1 ða [ CÞ: Gð1 2 aÞ

ð27Þ

Stirling functions of the first kind. Fractional Calculus

689

Proof. (a) Let ReðaÞ $ 0; a – 0, and n ¼ ½ReðaÞ% þ 1. By (6) and (7), sða; kÞ ¼

! "n $ 1 d # n2a lim I 0þ ½logðtÞ%k ðxÞ: k! x!1 dx

ð28Þ

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Now, using formula (2.44) of Samko et al. [16], # n2a g $ I 0þ t ðxÞ ¼

Gðg þ 1Þ x gþn2a ðg . 21Þ; Gðg þ 1 þ n 2 aÞ

whence a k-times differentiation with respect to g yields for g . 2 1, after an interchange of integration and differentiation, #

a g k I n2 0þ t ½logðtÞ%

$

a I n2 0þ

ðxÞ ¼ ¼

!

" ! ! "k › k g › # n2a g $ t ðxÞ ¼ I 0þ t ðxÞ ›g ›g

!

"% & › k Gðg þ 1Þ x gþn2a : ›g Gðg þ 1 þ n 2 aÞ

ð29Þ

Differentiating this expression k times with respect to x, interchanging the order of differentiation, one has for g . 2 1 ! "n ! "k % & $ d # n2a g › Gðg þ 1Þ k g2a x I 0þ t ½logðtÞ% ðxÞ ¼ : dx ›g Gðg þ 1 2 aÞ

ð30Þ

Thus for g . 21, ! "% & $ 1# a g 1 › k Gðg þ 1Þ k g2 a D t ½logðtÞ% ðxÞ ¼ x : k! 0þ k! ›g Gðg þ 1 2 aÞ Taking the limit for g ! 0, one has by (25) and (31) # $ # $ 1 1 lim Da ½logðtÞ%k ðxÞ ¼ lim lim Da0þ t g ½logðtÞ%k ðxÞ k! x!1 0þ k! x!1 g ! 0 ! "k % ! "k % & & 1 › Gðg þ 1Þx g2a 1 › Gðg þ 1Þ lim lim lim ¼ ¼ ; k! g ! 0 x!1 ›g k! g ! 0 ›g Gðg þ 1 2 aÞ Gðg þ 1 2 aÞ

sða; kÞ ¼

establishing part (a) for ReðaÞ $ 0; a – 0. If a ¼ 0, then by (6), # $ 1 1 lim D0 ½logðtÞ%k ðxÞ ¼ lim½logðxÞ%k ¼ sð0; kÞ ¼ Gðk þ 1Þ x!1 0þ k! x!1 Thus s(0, 0) ¼ 1, and s(0, k) ¼ 0 for k [ N.

(

1; for k ¼ 0

0; for k [ N:

ð31Þ

690

P. L. Butzer et al.

(b) In case Re(a) , 0 one applies (9), using arguments similar to the above, but using (6), (7) with n 2 a replaced by 2 a, giving ! " ! a g " 1 1 lim I 2a ½logðtÞ&k ðxÞ ¼ lim lim I 2 t ½logðtÞ&k ðxÞ k! x!1 0þ k! x!1 g ! 0 0þ # $k % & # $k % & 1 › Gðg þ 1Þ 1 › Gðg þ 1Þ lim lim lim x g2a ¼ ¼ : k! g ! 0 x!1 ›g Gðg þ 1 2 aÞ k! g ! 0 ›g Gðg þ 1 2 aÞ

sða; kÞ ¼

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When k ¼ 0, then, in accordance with (5), for any a [ C we have sða; 0Þ ¼

1 Gðx þ 1Þ 1 lim½xa & ¼ lim ¼ : x!0 Gðx þ 1 2 aÞ 0! x!0 Gð1 2 aÞ

This yields (27), and thus the theorem is proved. As a corollary of Theorem 2.1 and Theorem 1.2 ([17], p.107) we have Corollary 2.2 There holds, for a [ CnZ,

c ð1Þ 2 c ð1 2 aÞ FðaÞ ¼ ; Gð1 2 aÞ Gð1 2 aÞ

ð32Þ

D 1 FðaÞ þ F2 ðaÞ ; 2Gð1 2 aÞ

ð33Þ

D 2 FðaÞ þ 3FðaÞD 1 FðaÞ þ F3 ðaÞ ; 6Gð1 2 aÞ

ð34Þ

sða; 1Þ ¼

sða; 2Þ ¼ sða; 3Þ ¼ sða; 4Þ ¼

D 3 FðaÞ þ 4FðaÞD 2 FðaÞ þ 6F2 ðaÞD 1 FðaÞ þ 3½D 1 FðaÞ&2 þ 3F4 ðaÞ ; 24Gð1 2 aÞ

where D k FðaÞ U

# $k › ½c ðx þ 1Þ 2 c ðx þ 1 2 aÞ&jx¼0 ›x

ðk [ N0 Þ:

ð35Þ A

Proof. Concerning (32), recalling the proof of (14), (15), noting [0]a ¼ 1, % & d Gðx þ 1Þ c ð1Þ 2 c ð1 2 aÞ : sða; 1Þ ¼ lim ¼ lim½x &a Fðx; aÞ ¼ x!0 dx Gðx þ 1 2 aÞ x!0 Gð1 2 aÞ As this procedure is somewhat cumbersome for s(a, 2), etc. we apply the recursion formula (11). Indeed, sða; 2Þ ¼ 1=2{C1 ðaÞsða; 0Þ þ C0 ðaÞsða; 1Þ}, which is precisely (33). Let us also consider s(a, 3); in fact, 1 sða; 3Þ ¼ {C2 ðaÞsða; 0Þ þ C1 ðaÞsða; 1Þ þ C0 ðaÞsða; 2Þ}; 3 which turns out to be (34). The further formulas for s(a, k), k ¼ 4, 5, . . . , follow similarly. A

Stirling functions of the first kind. Fractional Calculus

691

2.2 Multiple sum or Euler Sum representations, Riemann zeta function First let us recall an expression of the classical s(k, m) in terms of a multiple sum; [18, 19]. First note that sðk; 1Þ ¼ ð21Þkþ1 ðk 2 1Þ!:

ð36Þ

For 2 # m # k one has

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sðk; mÞ ¼ ð21Þkþm ðk 2 1Þ! sðk; mÞ ¼ ð21Þ

kþm

k21 X

j¼m21

ðk 2 1Þ!

ð21Þ j sð j; m 2 1Þ

k21 X

jm21 ¼m21

!

ðm $ 1Þ; ð37Þ

" jm21 " X 21 ! j2 21 ! " X 1 1 ... : jm21 j ¼m22 jm22 j 1 j ¼1 1

m22

1

In particular, sðk; 2Þ ¼ ð21Þkþ2 ðk 2 1Þ!

k21 ! " X 1 ; j1 j ¼1 1

sðk; 3Þ ¼ ð21Þkþ3 ðk 2 1Þ!

21 ! " k21 ! "jX X 1 2 1 j2 ¼2

j2

j1 ¼1

j1

:

Our possibly new results in this respect are the sum representations of the following theorem and its corollary. Theorem 2.4.

For a [ C and m . ReðaÞ; m [ N, one has sða; mÞ ¼

1 1 X ð21Þk sðk; mÞ : Gð2aÞ k¼m k!ðk 2 aÞ

ð38Þ

Proof. Replacing z by 2 v in (2), dividing by v aþ1 and integrating this power series, using Abel’s limit theorem and Raabe’s convergence criterion, we obtain the relation ð 12 ð 1 1 X X ½logð1 2 vÞ&m dv ð21Þk sðk; mÞ 12 k212a ð21Þk sðk; mÞ : ¼ m! v dv ¼ m! 1þ a v k! k!ðk 2 aÞ 0þ 0þ k¼m k¼m Comparing this result with the integral representation (10) of s(a, m), we obtain immediately the result (38). Observe that (38) can be considered as the counterpart of a classical result of Stirling on the connection between his numbers s(k, m) and the famous Riemann zeta function

zðzÞ U

1 X j¼0

1 ð j þ 1Þz

ðReðzÞ . 1; z [ CÞ;

namely (see e.g. [21], p. 166, 195),

zðm þ 1Þ ¼

1 X ð21Þkþm sðk; mÞ : k!k k¼m

ð39Þ

Now we get to the Corollary mentioned, the multiple or Euler sum representation of s(a, m). A

692

P. L. Butzer et al.

For a [ C and m [ N, one has

Corollary 2.5.

sða; 1Þ ¼ 2

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sða; 2Þ ¼

1 1 X 1 ; Gð2aÞ k¼1 kðk 2 aÞ

ð40Þ

1 k21 X 1 X 1 1 ; Gð2aÞ k¼2 kðk 2 aÞ j¼1 j

ð41Þ

! j2 21 1 k21 X 1 X 1 1X 1 sða; 3Þ ¼ 2 ; ...; Gð2aÞ k¼3 kðk 2 aÞ j2 ¼2 j2 j1 ¼1 j1 sða; mÞ ¼

! " X " jX j2 21 ! " 1 ! m 21 ð21Þm X 1 1 1 ... : j Gð2aÞ jm ¼m jm ð jm 2 aÞ j ¼m21 jm21 1 j ¼1 m21

1

ð42Þ ð43Þ

Proof. In view of (36) and (38), sða; 1Þ ¼

1 1 X ð21Þk ð21Þkþ1 ðk 2 1Þ! ; Gð2aÞ k¼1 k!ðk 2 aÞ

ð44Þ

which is (40). Further, it also yields (41) since ( ) 1 k21 X 1 X ð21Þk 1 k sða; 2Þ ¼ : ð21Þ ðk 2 1Þ! Gð2aÞ k¼2 k!ðk 2 aÞ j j¼1

ð45Þ

Iterating this process yields the general multiple sum (43). The counterpart of the multiple sum formula (43) in the Zeta function setting, first established in Butzer et al. [18] (see Adamchik [22]), reads

zðm þ 1Þ ¼

1 ! " jX m 21 X 1

j2m

jm ¼m

jm21 ¼m21

!

1 jm21

"

···

j2 21 ! " X 1 ; j 1 j ¼1 1

ð46Þ

valid for any m . 0. The proof follows by inserting (37) into (39). The first multiple sum representation of z(2m) is also given in [18,19]. A closely related known multiple sum formula for z(m þ 1) is that of Mordell [23] of 1958, namely, for m [ N,

zðm þ 1Þ ¼

1 X 1 U T m: m! j ;· · ·j ¼1 j1 · · ·jm ð j1 þ · · · þ jm Þ 1

m

As to the proof Mordell noticed that Tm ¼

ð ð 1 X 1 u j1 þ· · ·þjm 21 ð21Þm 1 ½logð1 2 uÞ&m du ¼ zðm þ 1Þ; du ¼ m! j ; ... j ¼1 0 j1 . . . jm m! 0 u 1

m

the second part of this formula being in turn the counterpart of the representation (10) for s(a, m). A

Stirling functions of the first kind. Fractional Calculus

693

3. General Stirling functions s(a, b) with complex arguments, series representations In this section we will present some properties of the Stirling functions s(a, b), in particular their series representations.

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3.1 Preliminary lemmas The present section is devoted to the proof of Theorem 1.4 (a), (b), part (a) of which is a generalization of Theorem 1.1, the s(a, b) now being in their most general form, with a, b [ C. We will need the expansion ð1 2 zÞ2m ¼

1 X j¼0

ðmÞj

zj j!

ðjzj , 1; m [ CÞ;

ð47Þ

where z [ C, j [ N0 and (z)j is the Pochhammer symbol ([24], Section 2.1.1): ðzÞ0 U 1; ðzÞj U jð j þ 1Þ . . . ð j þ k 2 1Þ ðk [ NÞ:

ð48Þ

We also need three preliminary lemmas. For b [ C and m [ N0, there holds the relation ! "m m X › Gðb þ 1Þ cm; j ½logðtÞ&b2j ; ½logðxÞ þ logðtÞ&b ¼ lim x!1 ›x Gð b 2 j þ 1Þ j¼0

Lemma 3.1.

ð49Þ

where cm;m ¼ 1;

cm;0 ¼ 0 ðm [ NÞ;

ð50Þ

and cm; j ¼ cm21; j21 2 ðm 2 1Þcm21; j

ðm [ N;

j ¼ 1; . . . ; m 2 1Þ:

ð51Þ

In particular, c0;0 ¼ 1; c1;1 ¼ 1; c1;0 ¼ 0; c2;2 ¼ 1; c2;1 ¼ 21; c2;0 ¼ 0; c3;3 ¼ 1; c3;2 ¼ 23; c3;1 ¼ 2; c3;0 ¼ 0; cm; m21 ¼ 2

ðm 2 1Þm 2

cm;1 ¼ ð21Þm21 ðm 2 1Þ!;

ðm [ NÞ:

ð52Þ

Proof. For m ¼ 0 (49) is clear. If m ¼ 1 and m ¼ 2, then

› 1 ½logðxÞ þ logðtÞ&b ¼ b½logðxtÞ&b21 ; ›x x

ð53Þ

! "2 › bðb 2 1Þ b ½logðxtÞ&b ¼ ½logðxtÞ&b22 2 2 ½logðxtÞ&b21 : 2 ›x x x

ð54Þ

694

P. L. Butzer et al.

Taking the limit, as x ! 1, we have lim

›

x!1 ›x

½logðxtÞ$b ¼ b½logðtÞ$b21 ;

ð55Þ

! "2 › ½logðxtÞ$b ¼ bðb 2 1Þ½logðtÞ$b22 2 b½logðtÞ$b21 ; x!1 ›x

ð56Þ

lim

Downloaded by [Juan J. Trujillo] at 04:36 09 July 2011

and hence (49) follows for m ¼ 1 and m ¼ 2, respectively. There holds the following relation generalizing (53) and (54): xm

! "m m X › Gðb þ 1Þ cm; j ½logðxtÞ$b2j ; ½logðxtÞ$b ¼ ›x Gð b 2 j þ 1Þ j¼1

ð57Þ

where m [ N and cm, j are defined by (50) –(52). Then (49) will follow for m [ N0 from (57) by taking the limit, as x ! 1. Formula (57) is proved by induction. Indeed, it has the forms (53) and (54) for m ¼ 1 and m ¼ 2. Suppose that it is valid for m [ N. Using (57) we have x

mþ1

! "mþ1 › ½logðxtÞ$b ›x 8 9 < = m X › Gðb þ 1Þ cm; j ½logðxtÞ$b2j ¼ ; ›x : j¼1 Gðb 2 j þ 1Þ ¼

m m X X Gðb þ 1Þ Gðb þ 1Þ cm; j ½logðxtÞ$b2j21 2 m cm; j ½logðxtÞ$b2j Gð Gð b 2 jÞ b 2 j þ 1Þ j¼1 j¼0

¼

m X Gðb þ 1Þ Gðb þ 1Þ cm;m ½logðxtÞ$b2m21 þ cm; j21 ½logðxtÞ$b2j Gðb 2 m 2 1Þ Gð b 2 j þ 1Þ j¼2

2m

m X j¼2

Gðb þ 1Þ Gðb þ 1Þ cm; j ½logðxtÞ$b2j 2 m cm;1 ½logðxtÞ$b21 : Gðb 2 j þ 1Þ GðbÞ

By (50), noting c mþ1, m þ 1 ¼ 1, so cm;m ¼ 1 ¼ cmþ1;mþ1 ; c mþ1, mcm ¼ ð21Þmþ1 cmþ1;1 . Therefore, x mþ1

0

¼ 0; and

! "mþ1 › Gðb þ 1Þ cmþ1;mþ1 ½logðxtÞ$b2m21 ½logðxÞ þ logðtÞ$b ¼ ›x Gðb 2 m 2 1Þ þ

m X

þ

Gðb þ 1Þ cmþ1;1 ½logðxtÞ$b21 þ cmþ1;0 ½logðxtÞ$b : GðbÞ

j¼2

Gðb þ 1Þ ½cm; j21 2 mcm; j $½logðxtÞ$b2j Gðb 2 j þ 1Þ

This yields (57) with m being replaced by m þ 1, if we take (51) into account.

A

Stirling functions of the first kind. Fractional Calculus

Lemma 3.2.

695

Let a [ C, n [ N, k [ N0. Then ( 2 a 2 k)n has the representation ð2a 2 kÞn ¼

n X j¼0

ð21Þ j An; j ðk þ 1Þ j ;

ð58Þ

where the constants An; j ¼ An; j ðaÞ have the forms An;0 ¼ ð1 2 aÞð2 2 aÞ . . . ðn 2 aÞ ¼

Gðn þ 1 2 aÞ ; Gð1 2 aÞ

ð59Þ

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An;1 ¼ ð1 2 aÞð2 2 aÞ . . . ðn 2 1 2 aÞ þ ð1 2 aÞð3 2 aÞ . . . ðn 2 aÞ þ · · · þ ð2 2 aÞð3 2 aÞ . . . ðn 2 aÞ ¼

n Gðn þ 1 2 aÞ X 1 ; Gð1 2 aÞ j¼0 i 2 a

ð60Þ

An;2 ¼ ð1 2 aÞð2 2 aÞ . . . ðn 2 2 2 aÞ þ · · · þ ð3 2 aÞð4 2 aÞ . . . ðn 2 aÞ ¼

n Gðn þ 1 2 aÞ X 1 ; ... Gð1 2 aÞ i ;i ¼1 ði1 2 aÞði2 2 aÞ

ð61Þ

1 2 ði1 –i2 Þ

An; j ¼ ð1 2 aÞð2 2 aÞ . . . ðn 2 j 2 aÞ þ · · · þ ð j þ 1 2 aÞ . . . ðn 2 aÞ ¼

n n X Gðn þ 1 2 aÞ X 1 ¼ ðijþ1 2 aÞ· · ·ðin 2 aÞ; Gð1 2 aÞ i ; ... ;i ¼1 ði1 2 aÞ . . . ðij 2 aÞ i ; ... ;i ¼1 1

j ðik –ij Þ

jþ1

ð62Þ

n ðik –ij Þ

An;n22 ¼ ð1 2 aÞð2 2 aÞ þ ð1 2 aÞð3 2 aÞ þ · · · þ ð1 2 aÞðn 2 aÞ þ · · · þ ðn 2 1 2 aÞðn 2 aÞ ¼

n X

i1 ;i2 ¼1

ð63Þ

ði1 2 aÞði2 2 aÞ;

ði1 –i2 Þ

An;n21 ¼ ð1 2 aÞ þ ð2 2 aÞ þ · · · þ ðn 2 aÞ ¼

n X i¼1

ði 2 aÞ;

An;n ¼ 1:

ð64Þ ð65Þ

Proof. By (48) one has ð2a 2 kÞn ¼ ð2a 2 kÞð2a 2 k þ 1Þ . . . ð2a 2 k þ n 2 1Þ ¼ ¼

½ð1 2 aÞ 2 ðk þ 1Þ&½ð2 2 aÞ 2 ðk þ 1Þ& . . . ½ðn 2 aÞ 2 ðk þ 1Þ& n X j¼0

ð21Þ j An; j ðk þ 1Þ j ;

which yields (58). Since ( 2 a 2 k)n is a polynomial of degree n with respect to (k þ 1), then (59) – (65) follow from known results from algebra. A

696

P. L. Butzer et al.

Let n [ N0 and j ¼ 0, 1, . . . , n. There holds the following relations

Lemma 3.3.

n X m¼j

n m

!

Gðn þ 1 2 aÞ cm; j ¼ An; j Gðm þ 1 2 aÞ

ð j ¼ 0; 1; . . . ; nÞ;

ð66Þ

Downloaded by [Juan J. Trujillo] at 04:36 09 July 2011

where cm, j and An, j are given by (50) and (51) and (59) –(65), respectively. Proof. For j ¼ 0 or j ¼ n the proof of (66) is simple. If j ¼ 0, then in accordance with (52) c0, 0 ¼ 1, cm; j ¼ 0 ð j ¼ 1; . . . ; mÞ. Using such relations and (24), we have n X m¼0

n m

!

Gðn þ 1 2 aÞ Gðn þ 1 2 aÞ Gðn þ 1 2 aÞ cm;0 ¼ c0;0 ¼ ; Gðm þ 1 2 aÞ Gð1 2 aÞ Gð1 2 aÞ

ð67Þ

which proves (66) for j ¼ 0, if we take (59) into account. If j ¼ n, then (66) takes the form cn;n ¼ An;n , which is clear because according to (50) and (65) cn, n ¼ An, n ¼ 1. If j ¼ n 2 1, then by (24), and since cn21, n 2 1 ¼ 1, relation (66) takes the form nðn 2 aÞ þ cn;n21 ¼ An;n21

ðn [ NÞ:

ð68Þ

It is valid, since according to (52), nðn 2 aÞ þ cn;n21 ¼ ðnðn þ 1Þ=2Þ 2 na, while by (64), An;n21 ¼ ð1 þ 2 þ · · · þ nÞ 2 na ¼

nðn þ 1Þ 2 n a: 2

ð69Þ

The proofs of (66) in the cases j ¼ 1, . . . , n 2 2 can be carried out by direct applications of (50) – (52) and (59) – (65). They are cumbersome and therefore are omitted. A 3.2 Main theorem Now to Theorem 3.4, which was phrased in two parts in Theorem 1.4. for better understanding. Theorem 3.4.

Let a, b [ C such that Re(a) , Re(b). Then

1 1 ða þ 1Þj ebpi X ebpi X sða; bÞ ¼ ¼ ð21Þ j Gð2aÞ j¼0 j!ð j þ 1Þbþ1 Gð2aÞ j¼0

2a 2 1 j

!

1 ; ð j þ 1Þbþ1

ð70Þ

both series being absolutely convergent. Proof. We first establish the result for ReðaÞ , ReðbÞ; ReðbÞ $ 0. By definition (21), a change of variables u ¼ e2t, noting (16) and (47), an interchange of the order of

Downloaded by [Juan J. Trujillo] at 04:36 09 July 2011

Stirling functions of the first kind. Fractional Calculus

697

integration and summation yields that ð1 1 ½logðuÞ&b du sða; bÞ ¼ Gðb þ 1ÞGð2aÞ 0 ð1 2 uÞaþ1 ð1 1 ¼ ð1 2 e2t Þ2a21 ð2tÞb e2t dt Gðb þ 1ÞGð2aÞ 0 "ð 1 # ebpi lim ¼ ð1 2 e2t Þ2a21 t b e2t dt Gðb þ 1ÞGð2aÞ e !0þ e " # ð1 1 X ða þ 1Þj ebpi 2ð jþ1Þt b ¼ lim e t dt e !0þ e Gðb þ 1ÞGð2aÞ j¼1 j! " # 1 X ða þ 1Þj ebpi ¼ lim Gðb þ 1; e ð j þ 1ÞÞ ; Gðb þ 1ÞGð2aÞ j¼1 j!ð j þ 1Þbþ1 e !0þ Ð1 where Gðz; wÞ ¼ w t z21 e2t dt is the incomplete gamma function ([24], 6.9(21)). Since limw!0þ Gðz; wÞ ¼ GðzÞ, the last relation equals the first sum in (70). The second sum in (70) follows from the first by noting the property ! 2a 2 1 ða þ 1Þj j ¼ ð21Þ ð71Þ ða [ C; j [ N0 Þ: j j! As to the convergence of the two series in (70), consider the general term dj of the series ! 1 2a 2 1 X ebpi 1 ð21Þ j dj ; dj U : ð72Þ j Gð2 a Þ ð j þ 1Þbþ1 j¼0 In view of the estimate for binomial coefficients (24), namely % !% % a % c % % ða; b [ C; a – 21; 22; . . . Þ; %# % % b % b 1þReðaÞ

ð73Þ

for a certain constant c . 0, one has for dj the estimate jdj j #

c c ¼ : j ReðbÞþ1 j Reð2a21Þþ1 j ReðbÞþ12ReðaÞ

This estimate establishes the assertions of Theorem 3.4 only for Re(a) , 0 and Re(b) $ 0 since Re(b) þ 1 2 Re(a) . 1. If Re(a) $ 0, a – 0, and Re(b) . Re(a), then instead of applying the definition (21) we work with (19) and (8). Then s(a, b) takes on the form for n ¼ ½ReðaÞ& þ 1, & 'n ðx 1 › 1 lim ðx 2 uÞn2a21 ½logðuÞ&b du Gðb þ 1Þ x!1 ›x Gðn 2 aÞ 0 & 'n ð 1 1 › lim ¼ ð1 2 tÞn2a ½x n2a ½log xt &b &dt x!1 Gðb þ 1ÞGðn 2 aÞ ›x 0 & 'n ð1 1 › ¼ ð1 2 tÞn2a lim ½x n2a ½log xt &b &dt; x!1 Gðb þ 1ÞGðn 2 aÞ 0 ›x

sða; bÞ ¼

698

P. L. Butzer et al.

where the change of variables u ¼ xt was made. Applying now the Leibniz rule for the derivative of a product, taking into account

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! a g21 " GðgÞx g2a21 D0þ t ðxÞ ¼ Gðg 2 aÞ

ðReðaÞ . 0; ReðgÞ . 0Þ;

with a ¼ n 2 m and g ¼ n 2 a þ 1, as well as in Lemma 3.1, we find ! ð n 1 n X 1 sða; bÞ ¼ £ ð1 2 tÞn2a lim x!1 Gðb þ 1ÞGðn 2 aÞ m¼0 m 0 &$ %n2m ' $ %m d › £ ½x n2a & ½logðxÞ þ logðtÞ&b dt dx ›x ! n X n Gðn þ 1 2 aÞ 1 ¼ Gðb þ 1ÞGðn 2 aÞ m¼0 m Gðm þ 1 2 aÞ $ %m ð1 › n2a £ ð1 2 tÞ lim ½log xt &b dt x!1 › x 0 ! n n Gðn þ 1 2 aÞ X 1 ¼ Gðb þ 1Þ m¼0 m Gðm þ 1 2 aÞ ð1 m X Gðb þ 1Þ 1 cm; j £ ð1 2 tÞn2a ½logðtÞ&b2j dt: Gðb 2 j þ 1Þ Gðn 2 aÞ 0 j¼0 Hence we obtain the relation ! n m n Gðn þ 1 2 aÞ X X ! a " cm; j b2j ðxÞ: lim I n2 sða; bÞ ¼ 0þ ½logðtÞ& x!1 Gðm þ 1 2 a Þ Gð b 2 j þ 1Þ m m¼0 j¼0

ð74Þ

ð75Þ

ð76Þ

To evaluate the lim term for x ! 1, we apply the first part of the proof. Indeed, by (21) and (70) for 2 a and b replaced by n 2 a and b 2 j, respectively, we have since Re(a) , n, ! a " b2j ðxÞ ¼ Gðb 2 j þ 1Þsða 2 n; b 2 jÞ lim I n2 0þ ½logðtÞ& x!1

¼

1 eðb2jÞpi Gðb 2 j þ 1Þ X ða 2 n þ 1Þk : bþ12j Gðn 2 aÞ k¼0 k!ðk þ 1Þ

Thus (76) takes on the form n ebpi X sða; bÞ ¼ Gðn 2 aÞ m¼0

n m

!

m 1 X Gðn þ 1 2 aÞ X ða 2 n þ 1Þk ð21Þ j cm; j bþ12j Gðm þ 1 2 aÞ j¼0 k¼0 k!ðk þ 1Þ

1 n ebpi X ða 2 n þ 1Þk X ¼ b þ1 Gðn 2 aÞ k¼0 k!ðk þ 1Þ m¼0

n m

!

m Gðn þ 1 2 aÞ X ð21Þ j cm; j ðk þ 1Þ j Gðm þ 1 2 aÞ j¼0

1 n n X ebpi X ða 2 n þ 1Þk X j j ¼ ð21Þ ðk þ 1Þ Gðn 2 aÞ k¼0 k!ðk þ 1Þbþ1 j¼0 m¼j

n m

!

Gðn þ 1 2 aÞ cm; j : Gðm þ 1 2 aÞ ð77Þ

Stirling functions of the first kind. Fractional Calculus

699

According to Lemma 3.2 and Lemma 3.3, n X j¼0

j

ð21Þ ðk þ 1Þ

j

n X m¼j

0 @

n

1

A Gðn þ 1 2 aÞ cm; j ¼ m Gðm þ 1 2 aÞ

n X j¼0

ð21Þ j An; j ðk þ 1Þ j ¼ ð2a 2 kÞn ; ð78Þ

and hence from (77) we obtain

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sða; bÞ ¼ ebpi

1 X ða 2 n þ 1Þk ð2a 2 kÞn

Gðn 2 aÞ

k¼0

1 : k!ðk þ 1Þbþ1

ð79Þ

But ða 2 n þ 1Þk ð2a 2 kÞn =Gðn 2 aÞ ¼ ða þ 1Þk =Gð2aÞ, and thus (79) yields the first series in (70). P The second series in (70), clearly following from the first one, has the form 1 j¼0 dj of (72) with the dj term again having the estimate jdj j % cð jÞ2ðReðbÞþ12ReðaÞÞ ; thus it is convergent for ReðaÞ , ReðbÞ. The relations in (70) remain valid also for a ¼ 0; ReðbÞ . 0. Indeed, according to definition (20) for a ¼ 0, sð0; bÞ ¼ ð1=Gðb þ 1ÞÞ limx!1 ½log x&b ¼ 0, and " # 1 ða þ 1Þj ebpi X lim ð80Þ bþ1 ¼ 0: a!0þ Gð2aÞ j¼0 j!ð j þ 1Þ A

This completes the proof of Theorem 3.4.

It follows from Theorem 3.4 that if a; b [ C such that ReðaÞ , ReðbÞ, then the Stirling functions of the first kind s(a, b), defined by (19) –(21), have the same representations, namely sða; bÞ ¼

ebpi Gð2aÞ

1 P

j¼0

ðaþ1Þj j!ð jþ1Þbþ1

¼

ebpi Gð2aÞ

1 P

j¼0

ð21Þ

j

2a 2 1 j

!

1 ð jþ1Þbþ1

:

A As a corollary of Theorem 3.2 we have Corollary 3.5. (a) If n [ N0 and b [ C with Re(b) . n, then sðn; bÞ ¼ 0;

sð0; bÞ ¼ 0:

(b) If n [ N and b [ C such that ReðbÞ . 2n, then n21 n21 ebpi X ð21Þ j sð2n; bÞ ¼ j ðn 2 1Þ! j¼0 n n ebpi X ¼ ð21Þ j21 j n! j¼1

!

1 : jb

!

ð81Þ

1 ð j þ 1Þbþ1 ð82Þ

700

P. L. Butzer et al.

In particular, for b [ C, sð21; bÞ ¼ ebpi ;

! " 1 sð22; bÞ ¼ ebpi 1 2 bþ1 : 2

ð83Þ

Proof. As to equation (82), under the given conditions its left hand side exists, being given according to (21) with a ¼ 2 n by

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sð2n; bÞ ¼

1 Gðb þ 1Þðn 2 1Þ!

ð1 0

ð1 2 tÞn21 ½log t &b dt

ð84Þ

a convergent integral for Re(b) . 2 n. The right hand side turns out to be a finite sum since for a ¼ 2 n, ( 2 n þ 1)j ¼ 0 for j ¼ n, n þ 1, . . . , so that it exists for any n [ N and b [ C. To establish the right hand side of (82) one replaces j þ 1 by k, and observes that ! ! n21 1 n n ¼ ðn [ N; 1 % k % n 2 1Þ: ð85Þ k21 k k A 3.3 Differentiability of the s(a, b); the Zeta function encore In Section 2.2 we indicated that the Stirling functions s(a, m) are closely connected to the Zeta function z(m þ 1). This close connection is indeed true also for the most general s(a, b) with a, b [ C; the missing link is Theorem 3.4. For this purpose we first need a result on the continuity and differentiability of the s(a, b) with respect to a; the corresponding result for b is also given. Theorem 3.6. Let a, b [ C be complex numbers such that Re(a) , Re(b). Then there holds the following assertions: (a) s(a, b) as a function of a is continuously differentiable for a [ C, a – 0, and 1 ða þ 1Þj › ebpi X sða; bÞ ¼ ½c ð2aÞ 2 c ða þ 1Þ&sða; bÞ þ c ða þ 1 þ jÞ: ð86Þ ›a Gð2aÞ j¼0 j!ð j þ 1Þbþ1

(b) s(a, b) as function of b is continuously differentiable for b [ C, and for m [ N, 0 1 $ %m m 1 m ða þ 1Þj X › ebpi X k m2k @ A sða; bÞ ¼ ð21Þ ðipÞ ½logð j þ 1Þ&k : ð87Þ b þ1 ›b Gð2aÞ j¼0 j!ð j þ 1Þ k k¼0 Proof. The continuity of s(a, b) as functions of a and b follow from the first formula of (70). The relations (86), (87) are deduced by differentiation with respect to a and b, respectively. As to the former one makes use of the fact that d 1 c ð2aÞ ¼ ; dx Gð2aÞ Gð2aÞ

d ða þ 1Þj ¼ ða þ 1Þj ½c ða þ 1 þ jÞ 2 c ða þ 1Þ& da

Stirling functions of the first kind. Fractional Calculus

701

and, as to the latter, noting Leibniz’s rule, !

› ›b

"m # ebpi

$ X m 1 ebpi ð21Þk ðipÞm2k ¼ bþ1 ð j þ 1Þ ð j þ 1Þbþ1 k¼0

m k

!

½logð j þ 1Þ&k :

The convergence of the series on the right sides of (86) and (87) follows by applying the relations (71), (73) and the following asymptotic formulae for the gamma and psi-functions (see e.g. [24], 1.18 (4) and 1.18 (7)),

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Gðz þ aÞ ¼ z a2b ½1 þ Oðz 21 Þ&; Gðz þ bÞ

c ðzÞ ¼ logðzÞ þ Oðz 21 Þ ðz ! 1Þ:

ð88Þ

Finally to the new connection between s(a, b) and z(b þ 1), the particular case b ¼ m of which was first established in [4]. A Theorem 3.7.

Let a, b [ C such that Re(b) . 0 and Re(a) , Re(b). Then

lim Gð2aÞsða; bÞ ¼ ebpi zðb þ 1Þ;

a!0

lim

›

a!0 ›a

sða; bÞ ¼ 2ebpi zðb þ 1Þ:

ð89Þ

Proof. In view of Theorem 3.4 we have for Re(b) . 0 with Re(a) , Re(b), lim Gð2aÞsða; bÞ ¼ lim ebpi

a!0

a!0

1 X j¼0

ða þ 1Þj : j!ð j þ 1Þbþ1

Since (1)j ¼ j! for j [ N0, the latter limit is nothing but ebpiz(b þ 1), establishing the first formula of (89). As to the second formula of (89), note that, in accordance with [24], formula 1.7 (11), and [9], formula 6.1.3, one has # ! "$ 1 1 c ð2aÞ 2 c ða þ 1Þ ¼ p cotðapÞ; Gð2aÞ ¼ 2 1 þ O ð90Þ ða ! 0Þ: z a Since p cotðpaÞ , 1=a as a ! 0, then lim

a!0

c ð2aÞ 2 c ða þ 1Þ ¼ 21: Gð2aÞ

ð91Þ

Taking the limit as a ! 0 in (86), and using (90) and (91), we deduce the second formula of (89). Let us observe that this theorem gives another proof of the interesting but hardly known multiple sum representation (46) of the Zeta function. A

4. Recurrence relations for s(a, b) Theorem 4.1. Let a, b [ C such that Re(a) , Re(b) 2 1. Then the function s(a, b) satisfies the recurrence formula sða þ 1; bÞ ¼ sða; b 2 1Þ 2 asða; bÞ;

ð92Þ

702

P. L. Butzer et al.

and for n [ N0 in addition

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sða þ 1; bÞ ¼

n X j¼0

ð2aÞj sða 2 j; b 2 1Þ þ ð2aÞnþ1 sða 2 n; bÞ:

ð93Þ

Proof. If ReðaÞ , ReðbÞ 2 1, then according to Theorem 3.4 the Stirling functions s(a þ 1, b), s(a, b 2 1) and s(a, b) are well-defined. For a ¼ 0 relation (92) takes the form s(1, b) ¼ s(0, b 2 1) which is obvious since s(1, b) ¼ s(0, b 2 1) ¼ 0 by (81). Let a – 0. In view of Theorem 3.4 we have, since both series on the right-hand side of (92) are absolutely convergent, sða; b 2 1Þ 2 asða; bÞ ¼

1 1 ða þ 1Þj ða þ 1Þj eðb21Þpi X aebpi X 2 Gð2aÞ j¼0 j!ð j þ 1Þb Gð2aÞ j¼0 j!ð j þ 1Þbþ1

¼

1 ða þ 1Þj ebpi X ½2ða þ j þ 1Þ& Gð2aÞ j¼0 j!ð j þ 1Þbþ1

¼

1 ða þ 2Þj ebpi ð2a 2 1Þ X ; Gð2aÞ j!ð j þ 1Þbþ1 j¼0

where the relation (a þ 1)j(a þ 1 þ j) ¼ (a þ 1)(a þ 2)j, j [ N0, was used. This will yield (92). Relation (93) will be established by induction. Indeed, it coincides with (92) for n ¼ 0. Supposing it is valid for n [ N, then by (92), nþ1 X j¼0

ð2aÞj sða 2 j; b 2 1Þ ¼ sða þ 1; bÞ þ ð2aÞnþ1 ½sða 2 n 2 1; b 2 1Þ 2 sða 2 n; bÞ& ¼ sða þ 1; bÞ 2 ð2aÞnþ1 ða 2 n 2 1Þsða 2 n 2 1; bÞ

ð94Þ

which will take on the form nþ1 X j¼0

ð2aÞj sða 2 j; b 2 1Þ ¼ sða þ 1; bÞ þ ð2aÞnþ2 ða 2 n 2 1Þsða 2 n 2 1; bÞ;

noting that 2 ( 2 a)nþ1(a 2 n 2 1) ¼ ( 2 a)nþ2. This yields (93), n being replaced by n þ 1. Thus the proof is complete. A Theorem 4.2. For m [ N and b [ C the Stirling functions s( 2 m, b) satisfy the recurrence formula sð2m; bÞ ¼ sð2m 2 1; b 2 1Þ þ ðm þ 1Þsð2m 2 1; bÞ;

ð95Þ

and for n [ N0 in addition sð2m; bÞ ¼

n X j¼0

ðm þ 1Þj sð2m 2 1 2 j; b 2 1Þ þ ðm þ 1Þnþ1 sð2m 2 1 2 n; bÞ:

ð96Þ A

Stirling functions of the first kind. Fractional Calculus

703

Proof. In view of formula (92) we have sð2m 2 1; b 2 1Þþðm þ 1Þsð2m 2 1; bÞ

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0 1 0 1 m m m m bpi X eðb21Þpi X 1 e 1 j@ A ¼ ð21Þ j @ A þ ð21Þ b bþ1 m! j¼0 m! j ð j þ 1Þ j ð j þ 1Þ j¼0

0 1 " n ! m ebpi X mþ1 1 j@ A ¼ ð21Þ 2 m! j¼0 ð j þ 1Þbþ1 ð j þ 1Þb j 0 1 m m21 ebpi X j@ A ðm 2 jÞ : ¼ ð21Þ bþ1 m! j¼0 j ð j þ 1Þ

Since one has ! 1 m ðm 2 jÞ ¼ m j

m21 j

!

ð97Þ

ðm [ N; 1 # j # m 2 1Þ;

m21 m21 ebpi X ð21Þ j sð2m 2 1; b 2 1Þ þ ðm þ 1Þsð2m 2 1; bÞ ¼ j ðm 2 1Þ! j¼0

!

1 ; ð j þ 1Þbþ1 A

which yields (95). Relation (96) follows from (95) by induction. Corollary 4.3.

Let m and k both belong to N such that m . k. Then sð2m; 2kÞ ¼ 0 ðm; k [ N; m . kÞ:

ð98Þ

Proof. We apply (95) with b ¼ 1 2 k and m replaced by m 2 1, thus sð2m; 2kÞ ¼ sð1 2 m; 1 2 kÞ 2 msð2m; 1 2 kÞ:

ð99Þ

Now we proceed by induction. It holds for k ¼ 1 with m . 1, since s( 2 n, 0) ¼ 1/n!, s( 2 m, 21) ¼ s(1 2 m, 0) 2 ms( 2 n, 0) ¼ 1/(m 2 1)! 2 m/m! ¼ 0. Now suppose (98) holds for k ¼ n [ N, m . n, i.e. s( 2 m, 2n) ¼ 0. If m . n þ 1, then by (99) with k ¼ n þ 1, s( 2 m, 2n 2 1) ¼ s(1 2 m, 2n) 2 ms( 2 m, 2n) ¼ 0 (since s(1 2 m, 2n) ¼ s( 2 m, 2n) ¼ 0). This gives (98) with k ¼ n þ 1. A Example 1.

This corollary yields, in particular, the relations

sð22; 21Þ ¼ sð23; 21Þ ¼ sð23; 22Þ ¼ sð24; 21Þ ¼ sð24; 22Þ ¼ sð24; 23Þ ¼ 0: Remark 3.

Observe that for b ¼ k [ Z, Theorem 4.2 reduces to sða þ 1; kÞ ¼ sða; k 2 1Þ 2 asða; kÞ

ða [ C; ReðaÞ , k 2 1Þ:

ð100Þ

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P. L. Butzer et al.

However, according to Theorem 2.1 and results in [2,4] the latter formula is valid for any a [ R and any k [ N. A natural question is whether the formula does remain valid at least for a [ R and non-positive k [ Z2 0 with a $ k 2 1.

5. Connections between the Stirling functions of first and second kind

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In this section we prove connections between the Stirling functions of the first and second kind s(a, b) and S(a, k), the latter defined in (18). 5.1 Coincidence relations The Stirling functions of the first kind s(2 n, b), defined for n [ N0 and b [ C by (70) coincide, apart from a multiplicative factor, with the Stirling functions of the second kind S(2 b, n), defined in (18). The situation is similar for s(2 k, 2 a) and S(a, k). Theorem 5.1. (a) Let b [ C (b – 0) and n [ N. The Stirling functions s( 2 n, b) coincide with S( 2 b, n) apart from a constant multiplier: sð2n; bÞ ¼ ð21Þn e2ðbþ1Þp Sð2b; nÞ:

ð101Þ

In particular, for m [ Z, m – 0, sð2n; mÞ ¼ ð21Þnþm21 Sð2m; nÞ:

ð102Þ

(b) Let a [ C (a – 0) and k [ N. The Stirling functions S(a, k) coincide with the s( 2 k, 2 a) apart from a constant multiplier: Sða; kÞ ¼ ð21Þk eðaþ1Þp sð2k; 2aÞ:

ð103Þ

Sðm; kÞ ¼ ð21Þmþkþ1 sð2k; 2mÞ:

ð104Þ

In particular, for m [ Z,

A Proof. The result in (101) follows from Theorem 2 in [10], Theorem 7, if we take into account the explicit representations for s( 2 n, b) and S( 2 b, n) given by (70) and (18). (103) clearly follows from (101). The above theorem enables one to transfer several results we have established for the Stirling functions of second kind S(a, k) to such for the Stirling functions of first kind s( 2 n, b). First recall that the Liouville fractional derivative of order a [ C, Re(a) $ 0, is defined for x [ R, n ¼ [Re(a)] þ 1 by # $n ðx ! a " d 1 f ðtÞdt ðReðaÞ $ 0Þ: ð105Þ Dþ f ðxÞ U dx Gðn 2 aÞ 21 ðx 2 tÞa2nþ1

Stirling functions of the first kind. Fractional Calculus

705

One such result ([10], Theorem 8), namely that S(a, k) can be expressed in terms of this derivative in the form Sða; kÞ ¼

! " ð21Þk lim Daþ ½ð1 2 et Þk 2 1& ðxÞ k! x!0

for a [ C, Re(a) . 0, k [ N can be transferred to Theorem 5.2.

ð106Þ A

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Theorem 5.2. Let b [ C, Re(b) , 0, and n [ N. Then the Stirling functions s( 2 n, b) have the Liouville fractional integral representation sð2n; bÞ ¼

# $ eðbþ1Þpi b t k lim D2 þ ½ð1 2 e Þ 2 1& ðxÞ n! x!0

Proof. The result in (107) follows directly from (106) and (101). Remark 4.

ð107Þ

A

Formula (107) can be used as an alternative definition of s( 2 n, b).

5.2 Results from sampling analysis For the counterpart of the classical orthogonality property (see below) for the Stirling functions we need the sampling theorem of signal analysis. Let BppW , W . 0, 1 # p , 1, be the class of those functions g [ L p ðRÞ having an extension to the complex plane C as an entire function of exponential type pW, namely jgðzÞj # expðpWjyjÞkgkL p

ðz ¼ x þ iy; x; y [ RÞ:

ð108Þ

The sampling theorem now states: Theorem 5.3 (Sampling theorem). Any signal function g [ BppW , 1 # p , 1, some W . 0, can be completely reconstructed from its sampled values g( j/W) taken at the nodes j/W, j [ Z, in terms of % & 1 X j sin½p ðWz 2 jÞ& gðzÞ ¼ ; g W p ðWz 2 jÞ j¼21

ð109Þ

the series converging absolutely and uniformly on compact subsets of C. For literature regarding sampling analysis see e.g. Butzer – Splettstoesser – Stens [11], Butzer– Schmeisser – Stens [12], Butzer [13] and Higgins [14]. Further we shall need the known sampling representation of the binomial coefficient function (28), being the particular case of Theorem 2.1 in Ref. [5].

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Lemma 5.4. If a [ C and Re(z) . 2 1, then 1 # p , 1, and there holds z

a

!

¼

1 X k¼0

z k

!

z

a

!

as a function of a belongs to Bpp ,

sin½p ða 2 kÞ% ða [ CÞ: p ða 2 kÞ

ð110Þ

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5.3 Generalized orthogonality property The classical Stirling functions of the first and second kind s(n, k) and S(n, k) are connected by the basic orthogonality relation (see e.g. [6 – 8] or [9], p. 825) n X k¼m

sðn; kÞSðk; mÞ ¼

n X k¼m

Sðn; kÞsðk; mÞ ¼ dn;m

ðm; n [ N0 Þ;

ð111Þ

where dm, n ¼ 1 (m ¼ n), dm, n ¼ 0 (m – n), the Kronecker delta. A partial counterpart for the Stirling functions reads Theorem 5.5.

Let a [ C and m [ N0. Then 1 X k¼m

sða; kÞSðk; mÞ ¼

Gða þ 1Þ sin½p ða 2 mÞ% : Gðm þ 1Þ p ða 2 mÞ

ð112Þ

Proof. Basic for the proof of the relation (112) is the well known sampling theorem for s(a, k), with a [ C and k [ N0, namely 1 X sða; kÞ sðn; kÞ sin½ða 2 nÞp % ¼ Gða þ 1Þ n¼k n! ða 2 nÞp

ða [ C; k [ N0 Þ;

ð113Þ

(see e.g. ([5], Theorem 4.1)). Noting the property S(k, m) ¼ 0 (k, m [ N0; k , m), applying the left-hand side of (111), changing the orders of summation and observing the known property of s(n, k) (see e.g. [9], p. 168) s(n, k) ¼ 0 (n, k [ N0 for n , k), then from (113) we obtain the relation: 1 X k¼m

sða; kÞSðk; mÞ ¼

1 X k¼0

sða; kÞSðk; mÞ ¼ Gða þ 1Þ

¼ Gða þ 1Þ ¼ Gða þ 1Þ

1 X n X

sðn; kÞSðk; mÞ

n¼0 k¼0

" 1 n X X n¼m

1 X 1 X sðn; kÞ sin½p ða 2 nÞ% Sðk; mÞ n! p ða 2 nÞ k¼0 n¼k

k¼m

sin½p ða 2 nÞ% n!p ða 2 nÞ

sðn; kÞSðk; mÞ

#

sin½p ða 2 nÞ% : n!p ða 2 nÞ

Stirling functions of the first kind. Fractional Calculus

707

Using (111), we deduce 1 X

1 X sin½p ða 2 nÞ& Gða þ 1Þ sin½p ða 2 mÞ& ¼ ; sða; kÞSðk; mÞ ¼ Gða þ 1Þ dðn; mÞ n! m! p ð a 2 nÞ p ða 2 mÞ n¼m k¼0

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which proves (112) and the proof is complete. Observe that formula (112) for a ¼ n [ N reduces to the first half of the classical (111). A Remark 5. The second half of the orthogonality property (111) is as yet unsolved. It amounts to whether the assertion 1 X m¼k

Sða; kÞsðk; mÞ ¼

Gða þ 1Þ sin½p ða 2 mÞ& : Gðm þ 1Þ p ða 2 mÞ

ð114Þ

is valid for a [ C, m [ N0. A seemingly related result is the sampling theorem for the Stirling functions S(a, m), namely, 1 X Sða; mÞ Sðk; mÞ sin½ða 2 kÞp & ¼ Gða þ 1Þ k¼m k! ða 2 kÞp

ða [ C; m [ N0 Þ;

ð115Þ

conjectured some years ago. Whereas the sampling theorem is valid for the function s(a, m), formula (113) above, it is not valid for S(a, m). Since S(a, 1) ¼ 1 for a [ C, a – 0 (see formula (7.6) in [10]) it would imply that 1 X 1 1 sin½ða 2 kÞp & ¼ ða [ C; m [ N0 Þ; Gða þ 1Þ k¼m k! ða 2 kÞp

ð116Þ

the possible sampling theorem for ½Gða þ 1Þ&21 which can be shown to be false. We establish its incorrectness for m ¼ 1. In this case, using relation sin½ða 2 kÞp& ¼ ð21Þk sinðapÞ, we can present the right-hand side of (116) in terms of the Kummer confluent hypergeometric function F(a; c; z) (see e.g. [24], Section 6.1) and rewrite (116) with m ¼ 1 as ! " 1 sinðapÞ 1 1 ¼2 2 Fð2a; 1 2 a; 21Þ þ : ð117Þ Gða þ 1Þ p a a If Re(a) , 0, then Fð2a; 1 2 a; 21Þ is expressed via the incomplete gamma functions: Fð2a; 1 2 a; 21Þ ¼ 2ag ð2a; 1Þ ¼ 2a½Gð2aÞ 2 Gð2a; 1Þ&; [24], 6.9 (22) with a ¼ 2 a and x ¼ 1. By the first formula in Remark 2, 2sinðapÞ=p ¼ ½Gð2aÞGða þ 1Þ&21 ; and thus (117) takes the form ð1 1 Gð2a; 1Þ ; t 2a21 e2t dt ¼ ðReðaÞ , 0Þ: ð118Þ a 1 But this relation is false. For example, for a , 0, the left-hand side of (118) is positive, while the right-hand side of (118) is negative. Thus relation (116) is not correct for m ¼ 1. Numerical examples which were also carried out independently by Markus Brede with a different procedure confirm this.

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6. Applications of sampling analysis and the generalized orthogonality property In this section we establish a connection between the Riemann – Liouville and Hadamard fractional derivatives, and represent a generalized fractional difference in terms of operators of the calculus of finite differences by means of the s(a, k). We also consider representative examples.

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6.1 The s(a, k) connecting two types of fractional derivatives; the operator d 5 xd/dx; applications One of the major applications of our joint paper ([10], Theorem 17) was the following: Theorem 6.1. Let f(x), x . 0, be an arbitrarily often differentiable function such that its Taylor series converges, and let a [ C. If Re(a) $ 0, the Hadamard fractional derivative Da0þ f , given by # $ ðx & ! a " d n 1 x'n2a21 du ð119Þ log f ðuÞ ; D0þ f ðxÞ U x dx Gðn 2 aÞ 0 u u with (n ¼ [Re(a)] þ 1) and x . 0, has the following representation !

1 X " Sða; kÞx k f ðkÞ ðxÞ: Da0þ f ðxÞ ¼

ð120Þ

k¼0

Our new result here is the inversion of the sum formula (120), namely 1 X ! " ! " sða; kÞ Dk0þ f ðxÞ x a Da0þ f ðxÞ ¼ k¼0

ða [ C; ReðaÞ $ 0Þ;

ð121Þ

expressing the classical Riemann –Liouville fractional derivative Da0þ f , given by (7), in terms of the Hadamard derivatives. As to (120), the Hadamard fractional derivative Da0þ f is expressed in terms of x k f ðkÞ ðxÞ. Observe that if a ¼ n [ N0, then (120) and (121) are the classical formulae (e.g. see [25], Lemma 9), respectively, !

n X " Dn0þ f ðxÞ ; ðd n f ÞðxÞ ¼ Sðn; kÞx k f ðkÞ ðxÞ k¼0

ðn [ N0 Þ;

ð122Þ

where d U (x(d/dx)), d n ¼ (x(d/dx))n, and # $n n X d xn f ðxÞ ¼ sðn; kÞd k f ðxÞ ðn [ N0 Þ: ð123Þ dx k¼0 ! " We shall now use a unified notation Da0þ f ðxÞ (a [ C) for the Riemann – Liouville fractional derivative (7) of order a and for the Riemann – Liouville fractional integral (8) of order 2 a in the cases Re(a) $ 0 and Re(a) , 0: Da0þ f U Da0þ f ðReðaÞ $ 0Þ; Now our result, the inversion of (120), reads

a Da0þ f U I 2 0þ f ðReðaÞ , 0Þ:

ð124Þ

Stirling functions of the first kind. Fractional Calculus

709

Theorem 6.2. Let f(x) be an arbitrarily often differentiable function on x . 0 such that x a ðDa0þ f ÞðxÞ=Gða þ 1Þ as a function of a [ C belongs to the class Bpp for 1 # p , ! 1. Then " there holds the expansion (121) for the Riemann – Liouville fractional derivative Da0þ f ðxÞ of order a [ C, Re(a) $ 0, provided that the series in the right-hand side of (121) converges.

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Proof. Inserting (122) (with n ¼ k) into the right-hand side of (121) and changing the orders of summations and applying (112), we have 1 X k¼0

1 k X X ! " sða; kÞ Dk0þ f ðxÞ ¼ sða; kÞ Sðk; jÞx j f ð jÞ ðxÞ j¼0

k¼0

¼

1 1 X X j¼0

k¼j

¼ Gða þ 1Þ

!

sða; kÞSðk; jÞ x j f ð jÞ ðxÞ

1 X sin½ða 2 jÞp & x j f ð jÞ ðxÞ : ða 2 jÞp j! j¼0

ð125Þ

By the assumption of the theorem, for x . 0, x z Dz0þ f =Gðz þ 1Þ as a function of z [ C belongs to the class Bpp . Applying the sampling formula (109) (with z ¼ a and W ¼ 1), and taking into account that ½Gð j þ 1Þ&21 ¼ 0 for j [ Z2, noting ðD j f ÞðxÞ ; ðDj0þ f ÞðxÞ ¼ f ð jÞ ðxÞ ( j [ N0), we deduce for a [ C ! " 1 x a Da0þ f ðxÞ X x j f ð jÞ ðxÞ sin½p ða 2 jÞ& ¼ : Gða þ 1Þ j! p ða 2 jÞ j¼0

ð126Þ

When a [ C, Re(a) $ 0, then (126) in accordance with (144) yields ! " 1 x a Da0þ f ðxÞ X x j f ð jÞ ðxÞ sin½p ða 2 jÞ& ¼ : Gða þ 1Þ j! p ða 2 jÞ j¼0 Thus (121) follows from (125) and (127), and the theorem is proved.

ð127Þ A

Corollary 6.3. If the conditions of Theorem 6.2 are satisfied, then for the integral I a0þ f of order a [ C, Re(a) . 0, one has the following sampling formula: ! " 1 x 2a I a0þ f ðxÞ X x j f ð jÞ ðxÞ sin½p ða þ jÞ& ¼ : Gð1 2 aÞ j! p ða þ jÞ j¼0

ð128Þ

To apply Theorem 6.2 we need the following auxiliary result Lemma 6.4. !Let l [ "C, ReðlÞ , 1 2 ð1=pÞ some" p . 1, and let Da0þ f be given by ! a for a 2l a a 2l ðxÞ=Gða þ 1Þ, x D0þ t logðtÞ ðxÞ=Gða þ 1Þ as functions of a [ C (144). Then x D0þ t

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are of exponential type p, ! " ! " x a Da0þ t 2l ðxÞ x a Da0þ t 2l logðtÞ ðxÞ [ Bpp ; [ Bpp : Gða þ 1Þ Gða þ 1Þ

ð129Þ

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In particular, x a ðDa0þ logðtÞÞðxÞ=Gða þ 1Þ [ Bpp . Proof. Let a [ C, l [ C, Re(l) , 0. By (74) and a similar formula for the Riemann – Liouville fractional integral (see e.g. [16], formula (2.44)) ! " x a Da0þ t 2l ðxÞ Gð1 2 lÞx 2l ¼ ðx . 0; ReðlÞ , 1Þ: ð130Þ Gða þ 1Þ Gð1 2 l 2 aÞGða þ 1Þ Set g1 ðzÞ ¼ Gð1 2 lÞt 2l ½Gð1 2 l 2 zÞGðz þ 1Þ&21 , where z ¼ x þ iy ¼ jzjei arg z ðx; y [ RÞ with 0 # arg(z) , 2p and t . 0 is a fixed positive number. g1(z) is clearly an entire function of z [ C. Noting G(z þ 1) ¼ zG(z) and the asymptotic relation GðzÞ , ð2pÞ1=2 e2z eðz21=2ÞlogðzÞ

ðz ! 1Þ

for the Gamma function ([24], formulas 1.2 (2) and 1.18 (2)), we have # $ jGð1 2 lÞjt 2ReðlÞ ey½argðzÞ2argð2z2lÞ& ðjzj ! 1Þ: jg1 ðzÞj , 12ReðlÞ 2peImðlÞargð2z2lÞ jzj

ð131Þ

ð132Þ

If jzj ! 1, then arg(z þ l) ! arg(z) and hence y½argðzÞ 2 argð2z 2 lÞ& # pjyj for sufficiently large jzj. Thus if we choose R . 0 sufficiently large, then relation (132) yields the estimate jg1 ðzÞj # Aepjyj

ðA . 0; jzj $ RÞ:

ð133Þ

ðB . 0; jzj # RÞ;

ð134Þ

If jzj # R, then a similar estimate holds jg1 ðzÞj # Bepjyj

because g1(z) is an entire function. When z ¼ x [ R and l [ C, Re(l) , 1 2 (1/p) for some p . 1, then in accordance with (132) g [ L p(R). Thus the side of (130) as a ! right-hand " function of a [ C is of exponential type p, and hence x a Da0þ t 2l ðxÞ=Gða þ 1Þ [ Bpp . By the known formula for the Riemann – Liouville fractional integral ([16], formula (2.50)), and a similar formula for the Riemann – Liouville fractional derivative, one has for x . 0 and Re(l) , 1, ! " x a Da0þ t 2l logðtÞ ðxÞ Gð1 2 lÞx 2l ¼ ½logðxÞ þ c ð1 2 lÞ 2 c ð1 2 l 2 aÞ&: ð135Þ Gða þ 1Þ Gð1 2 l 2 aÞGða þ 1Þ Set g2 ðzÞ ¼ Gð1 2 lÞt 2l ½logðtÞ þ c ð1 2 lÞ 2 c ð1 2 l 2 zÞ&½Gð1 2 l 2 zÞGðz þ 1Þ&21 where, as earlier, z ¼ x þ iy ¼ jzjei arg z (x, y [ R) with 0 # arg(z) , 2p and t . 0 is a fixed positive number. The functions G(1 2 l 2 z) and c(1 2 l 2 z) have the same simple poles zk ¼ 1 2 l þ k (k [ N0); see e.g. [24], p. 2 and p. 18. Therefore g2(z) is an entire function of z. Using the asymptotic relation (132) and the asymptotic estimate for c(z) at infinity,

Stirling functions of the first kind. Fractional Calculus

given in the second formula of (88), there holds the asymptotic relation ! " jGð1 2 lÞjt 2ReðlÞ ey½argðzÞ2argð2z2lÞ$ logjzj jg2 ðzÞj , ðjzj ! 1Þ: 12ReðlÞ 2pe2ImðlÞargð2z2lÞ jzj

711

ð136Þ

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Using the# same arguments $ as above, we deduce that g2(z) is of exponential # type p,$ and hence x a Da0þ t 2l logðtÞ ðxÞ=Gða þ 1Þ [ Bpp . In particular, when l ¼ 0, x a Da0þ logðtÞ ðxÞ= Gða þ 1Þ [ Bpp , and the lemma is proved. A Remark 6. By (130), the first result of Lemma 6.4 follows also from the sampling theorem for the binomial coefficients; see Lemma 5.4 with z ¼ 2 l in Section 5.2. Example 2. Here we consider an application of Theorem 6.2, namely to the power function f 1 ðxÞ ¼ x 2l (x . 0,l [ C). Then dx 2l ¼ 2lx 2l and d n x 2l ¼ ð2lÞn x 2l ¼ ðDn0þ t 2l ÞðxÞ for n . 1. But % &k d # 2l $ x ð137Þ ¼ ½2l $k x 2l2k ðl [ C; k [ N0 Þ; dx

(see e.g. [21], p. 196). This is confirmed by (123), giving for x . 0 and l [ C % &n n d # 2l $ X ¼ xn x sðn; kÞð2lÞk x 2l d k x 2l ¼ x 2l ½2l $n : dx k¼0

ð138Þ

By Theorem 6.2 and Lemma 6.4 the fractional version (121) reads for f1(x), with x . 0 and l [ C, ReðlÞ , 1 2 ð1=pÞ for some p . 1, jlj , 1, 1 1 X X # $ sða; kÞd k x 2l ¼ x 2l sða; kÞð2lÞk ¼ x 2l ½2l $a : x a Da0þ t 2l ðxÞ ¼ k¼0

k¼0

Note that series above converge for jlj , 1, and (139) gives # a 2l $ D0þ t ðxÞ ¼ ½2l $a x 2l2a ðx . 0; ReðlÞ , 1Þ;

ð139Þ

ð140Þ

which coincides with (130) (for Re(a) $ 0). The first one is the known sampling representation of the binomial coefficient function (24), being the particular case of Theorem 2.1 in [5]. Example 3. As a second application of Theorem 6.2, take f2(x) ¼ log(x) (x . 0). Then for d ¼ x(d/dx), d log(x) ¼ 1 and d mlog(x) ¼ 0 for m . 1. Then from (123) we have for n [ N % &n n X d xn logðxÞ ¼ sðn; kÞd k logðxÞ ¼ sðn; 0Þ logðxÞ þ sðn; 1Þ: dx k¼0 According to (27), (36) and the relation ½Gð j þ 1Þ$21 ¼ 0 for j [ Z2 , this yields for x . 0 and n [ N, % &n d 1 logðxÞ þ ð21Þn21 ðn 2 1Þ! ¼ ð21Þn21 GðnÞ: xn logðxÞ ¼ ð141Þ dx Gð1 2 nÞ

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P. L. Butzer et al.

By Theorem 6.2 and Lemma 6.4 and (27) and (32), the corresponding fractional version (121) for f2(x) ¼ log(x) reads For a [ C and x . 0, one has

Corollary 6.5.

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! " logðxÞ þ c ð1Þ 2 c ð1 2 aÞ : x a Da0þ logðtÞ ðxÞ ¼ sða; 0ÞlogðxÞ þ sða; 1Þ ¼ Gð1 2 aÞ

ð142Þ

Observe that for a ¼ n [ N (142) coincides with (141): # $n % ! " & d logðxÞ þ c ð1Þ 2 c ð1 2 aÞ xn logðxÞ ¼ lim x a Da0þ logðtÞ ðxÞ ¼ lim a!n a!n dx Gð1 2 aÞ ¼ 2 lim

a!n

c 0 ð1 2 aÞ ¼ ð21Þn21 GðnÞ: G0 ð1 2 aÞ

Example 4. As a third application of Theorem 6.2, take f 3 ðxÞ ¼ elx , l [ C, x . 0. Then for d ¼ x(d/dx), (122) and (123) take the respective forms !

n " ! " X Sðn; kÞðlxÞk elx Dn0þ elt ðxÞ ; d n elx ¼ k¼0

ðn [ N0 Þ;

ð143Þ

and xn

# $n n X d e lx ¼ sðn; kÞd k ðelx Þ ðn [ N0 Þ: dx k¼0

ð144Þ

The fractional version (121) of (144) is based on the following auxiliary result similar to Lemma 6.4. Lemma 6.6. type p,

! " For l [ C, x a Da0þ elt ðxÞ=Gða þ 1Þ as a function of a [ C is of exponential ! " x a Da0þ elt ðxÞ [ Bpp Gða þ 1Þ

for

1 # p , 1:

Proof. For a [ C there holds the following formula ! a lt " D0þ e ðxÞ ¼ x 2a E1;12a ðlxÞ

ða [ CÞ;

ð145Þ

ð146Þ

in terms of the the Mittag –Leffler function Em,b(z) defined for complex m, b [ C (Re(m . 0) by (see e.g. [26], Section 18.1; [27], Chapter III and [28]) Em;b ðzÞ ¼

1 X k¼0

zk Gðmk þ bÞ

ðz [ CÞ:

ð147Þ

Indeed, if Re(a) , 0, then by the second formula in (144) and Formula 8 in table 9.1 of [16] ! a lt " ! a lt " ðxÞ ¼ x 2a E1;12a ðlxÞ ðReðaÞ , 0Þ; D0þ e ðxÞ ¼ I 2 0þ e

Stirling functions of the first kind. Fractional Calculus

713

which proves (146) for Re(a) , 0. If Re(a) $ 0, then (146) is proved by using the first formula in (144), (7), Formula 8 in table 9.1 of [16] with a replaced by n 2 a, n ¼ [Re(a)] þ 1, and taking term by term differentiation. Now using (146), (147) and noting (24), we have ! ! " 1 1 k X x a Da0þ elt ðxÞ 1 ðlxÞk 1X ¼ ¼ ð148Þ ðlxÞk : Gða þ 1Þ k¼0 Gðk þ 1 2 aÞ k! k¼0 a Gða þ 1Þ k

[ Bpp , 1 # p , 1, for any k [ N0 , and hence ðx a ðDa0þ elt ÞðxÞÞ= a ðGða þ 1ÞÞ [ Bpp , which completes the proof of the lemma. Now, by Theorem 6.2 and Lemma 6.6, we deduce the fractional version (121) for f 3 ðxÞ ¼ elx , with x . 0 and l [ C. A By Lemma 5.4,

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!

Corollary 6.7.

For a [ C, l [ C and x . 0, one has ! " 1 x a Da0þ elt ðxÞ X 1 sin½p ða 2 kÞ& ðlxÞk elx ¼ ; Gða þ 1Þ Gðk þ 1Þ p ða 2 kÞ k¼0

ð149Þ

and the series on the right-hand side of (149) converges absolutely. # $ Proof. By Lemma 6.6 x a Da0þ elt ÞðxÞ=Gða þ 1Þ [ Bpp for 1 # !p , 1 "and we can apply Theorem 6.2 for the Riemann –Liouville fractional derivative Da0þ elt ðxÞ. Using (121), taking into account (143) and applying the generalised orthogonlity property (112), we have 1 1 k X X X ! " x a Da0þ elt ðxÞ ¼ sða; kÞd k ðelx Þ ¼ sða; kÞ Sðk; mÞðlxÞm elx k¼0

¼

1 X m¼0

k¼0

m lx

ðlxÞ e

"

1 X k¼m

m¼0

#

sða; kÞSðk; mÞ ¼

1 X m¼0

ðlxÞm elx

Gða þ 1Þ sin½p ða 2 kÞ& ; Gðm þ 1Þ p ða 2 kÞ

This yields (149). It is clear that the series on the right-hand side of (149) is absolutely convergent, and thus the corollary is proved. A Remark 7.

By (146), formula (149) with l [ C and x . 0 is equivalent to 1 E1;12a ðlxÞ X 1 sin½p ða 2 kÞ& ðlxÞk elx ¼ ða [ CÞ: Gða þ 1Þ Gðk þ 1Þ p ða 2 kÞ k¼0

ð150Þ

Since ½Gðk þ 1Þ&21 ¼ 0 for k [ Z2 ,! relations " (149) and (150) present the sampling representation (109) of gðaÞ ¼ x a Da0þ elt ðxÞ=Gða þ 1Þ ; E1;12a ðlxÞ=Gða þ 1Þ as a function of a [ C with parameters l [ C, x . 0. Note that the Mittag – Leffler function Em, b(z), which as a function of z is entire of order 1/m and type 1, plays an important role in fractional differential equations and in sampling analysis; [29], Chapter 4 and [30], respectively. Above it is considered as a function of b.

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6.2 The representation of a general fractional difference operator via s(a, k); the operator Q 5 xD; applications Let Dk be the finite difference of order k [ N0 given for x [ R by

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! k k X ! " Dk f ðxÞ ¼ D Dk21 f ðxÞ ¼ ð21Þk2j f ðx þ jÞ j j¼0

ðk [ N0 Þ

ð151Þ

with Df ðxÞ U D0 f ðxÞ ¼ f ðxÞ, D1 f ðxÞ ¼ f ðx þ 1Þ 2 f ðxÞ. A well-known operator in the calculus of finite differences, the operations of which are analogues to those of d ¼ xd/dx, is the operator u f ðxÞ U xDf ðxÞ, for which one has the iterative formula (see [21], p. 200)

u n f ðxÞ ¼

n X k¼1

½x þ k 2 1&k Sðn; kÞDk f ðxÞ ðn [ NÞ;

ð152Þ

where S(n, k) (n [ N, k ¼ 1, 2, . . . , n) are the Stirling numbers of second kind defined by the first relation in (18) for a ¼ n. If we multiply equation (152) by the Stirling functions of first kind s(m, n), sum it from n ¼ 1 to n ¼ m [ N and use the first orthogonality formula in (111), we obtain the inversion of the operator u n in the form (see e.g. [21], p. 200) Dm f ðxÞ ¼

m X 1 sðm; nÞu n f ðxÞ ½x þ m 2 1&m n¼1

ðm [ NÞ:

ð153Þ

We now establish a generalization of this relation for a generalized “infinite” or fractional order difference Daf, with complex order a [ C. For a [ C, Re (a) . 0, such a function is defined for suitable functions f by a genuine series ! 1 a X a api j D f ðxÞ U e ð21Þ ð154Þ f ðx þ jÞ ða [ C; ReðaÞ . 0Þ: j j¼0 Then Daf, a [ C, is defined by analytic continuation of Daf from Re(a) . 0 to a ! [ C. k 2 Note that if a ¼ k [ N0 , then, by (24) and 1/G( j þ 1) ¼ 0 ð j [ Z Þ, ¼ 0 for j j ¼ k þ 1, k þ 2, . . . , so that (154) turns out to be the classical finite difference Dk f given by (151). We now give a generalization of (153) in terms of a weighted generalization of (151) in the form

uvn f ðxÞ U

n X k¼0

½x þ k 2 1&k Sðn; kÞvðkÞDk f ðxÞ

ðn [ N0 Þ;

ð155Þ

where the weight v(a) is a function of a [ C. In particular, if n ¼ 0, then

uv0 f ðxÞ ¼ vð0Þ f ðxÞ;

ð156Þ

Stirling functions of the first kind. Fractional Calculus

715

and for v(a) ¼ 1 and n [ N, (155) coincides with (152):

u1n f ðxÞ ¼ u n f ðxÞ ðn [ NÞ;

ð157Þ

since ½x 2 1%0 ¼ 1 and S(0, 0) ¼ 1, S(n, 0) ¼ 0 (n [ N) (see p.169 of [21]). Theorem 6.8.

Let v(a) (a [ C) and f(x) (x [ R) be functions such that

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gva ð f Þ U

½x þ a 2 1%a vðaÞDa f ðxÞ [ Bpp Gða þ 1Þ

for some 1 # p , 1:

ð158Þ

Then Daf(x) has the representation, a generalization of (153), Da f ðxÞ ¼

1 X 1 sða; nÞuvn f ðxÞ; vðaÞ½x þ a 2 1%a n¼0

ð159Þ

if this series converges. Proof. Using (155) and interchanging the orders of summation, we have 1 X n¼0

sða; nÞuvn f ðxÞ ¼ ¼

1 X

sða; nÞ

n¼0

k¼0

1 1 X X k¼0

n X

n¼k

½x þ k 2 1%k Sðn; kÞvðkÞDk f ðxÞ !

sða; nÞSðn; kÞ vðkÞ½x þ k 2 1%k Dk f ðxÞ:

Then an application of the orthogonality property (112) yields 1 X n¼0

sða; nÞuvn f ðxÞ ¼

1 X Gða þ 1Þ sin½p ða 2 kÞ% vðkÞ½x þ k 2 1%k Dk f ðxÞ: Gðk þ 1Þ p ða 2 kÞ k¼0

ð160Þ

Now we apply Theorem 5.3 to gðzÞ ¼ gvz ð f Þ (with z ¼ a). We have 1 X ½x þ a 2 1%a ½x þ k 2 1%k sin½p ða 2 kÞ% ; vðaÞDa f ðxÞ ¼ vðkÞDk f ðxÞ Gða þ 1Þ Gðk þ 1Þ p ða 2 kÞ k¼21

which, since ½x þ k 2 1%k =Gðk þ 1Þ ¼ 0 for k [ Z2 , yields 1 X ½x þ a 2 1%a ½x þ k 2 1%k sin½p ða 2 kÞ% : vðaÞDa f ðxÞ ¼ vðkÞDk f ðxÞ Gða þ 1Þ Gðk þ 1Þ p ða 2 kÞ k¼0

Now (159) follows from (160) and (161), and thus the theorem is proved.

ð161Þ A

Corollary 6.9. If the conditions of Theorem 6.8 are satisfied, then for the generalized fractional difference Da f of order a [ C, Re(a) $ 0, there holds the series representation (159) provided this series converges.

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Remark 8. When v(a) ¼ 1 and a ¼ m [ N, relation (159) coincides with (153). Indeed, according to (27) with a ¼ m and (81) with n ¼ m, b ¼ n, sðm; 0Þ ¼ 0 ðm [ NÞ;

sðm; nÞ ¼ 0 ðm [ N0 ; n [ N; m , nÞ:

ð162Þ

Thus using the definition of Da and (157), we have for a ¼ m [ N

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Dm f ðxÞ ¼ Dm f ðxÞ ¼

1 m X X 1 1 sðm; nÞu1n f ðxÞ ¼ sðm; nÞu n f ðxÞ: ½x þ m 2 1&m n¼0 ½x þ m 2 1&m n¼1

To apply Theorem 6.8 we need a auxiliary result which yields explicit expressions for the fractional order differences (154) of two functions. Lemma 6.10.

Let a, l [ C, Re(a) . 0. Then there hold the following relations

D

a

!

Da ðelx Þ ¼ eapi elx ð1 2 el Þa ðl , 0Þ;

ð163Þ

"

ð164Þ

1 ½x 2 1&l

¼ eapi

Gðx 2 lÞGða þ lÞ GðlÞGðx þ aÞ

ðReða þ lÞ . 0Þ:

Proof. First note that in accordance with (154) and (73) the infinite series Da ðelx Þ converges for any a [ C (Re(a) . 0) and l , 0. Using the formula

ð21Þ

j

a j

!

¼

ð2aÞj j!

ða [ C; j [ N0 Þ;

ð165Þ

and (47) (with z ¼ el and m ¼ 2 a), we have for (163) Da ðelx Þ ¼ eapi

1 X ð2aÞj lðxþjÞ e ¼ eapi elx ð1 2 el Þa : j! j¼0

Using the first asymptotic estimate in (88), and taking (73) into account, we have for a [ C the estimate # ! "## # a ! 1 1 # # j # # B ReðaþlÞþ1 #ð21Þ j # ½x þ j 2 1&l # j

ðB . 0; j [ NÞ:

ð166Þ

Thus the series Da ð1=½x 2 1&l Þ converges when Re(a) þ Re(l) . 0. Applying the relation Gðz þ kÞ ¼ ðzÞk GðzÞ ðz [ C; k [ N0 Þ and the formula for the Gauss hypergeometric function (see, Erdelyi et al. [24], 2.1(14)): 2 F 1 ½a; b; c; 1&

¼

GðcÞGðc 2 a 2 bÞ Gðc 2 aÞGðc 2 bÞ

ðReðc 2 a 2 bÞ . 0Þ;

ð167Þ

Stirling functions of the first kind. Fractional Calculus

717

and taking (167) into account, we obtain ! " 1 1 X ð2aÞj Gðx 2 l þ jÞ ð2aÞj ðx 2 lÞj 1 Gðx 2 lÞ X a ¼ eapi D ¼ eapi ½x 2 1"l Gðx þ jÞ GðxÞ j! j! ðxÞj j¼0 j¼0 ¼ eapi

Gðx 2 lÞ Gðx 2 lÞGða þ lÞ ; F 1 ½2a; x 2 l; x; 1" ¼ eapi GðxÞ 2 GðlÞGðx þ aÞ

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which proves (164). This complete the proof of the lemma. By Lemma 6.10, the corresponding analytic continuations Da ð f Þ for a [ C have the same forms Da ðelx Þ ¼ eapi elx ð1 2 el Þa ðl [ CÞ; Da

!

1 ½x 2 1"l

"

¼ eapi

Gðx 2 lÞGða þ lÞ ðl [ CÞ: GðlÞGðx þ aÞ

ð168Þ ð169Þ

A Example 5. Here we consider an application of Theorem 6.8, namely to the factorial function f 4 ðxÞ U 1=½x 2 1"l with complex x; l [ C. For this specific function, the operator u U xD, which plays an important role in the calculus of finite differences, has its repeated operations given by (see e.g. [21], p. 200)

u k f 4 ðxÞ ¼

ð2lÞk ½x 2 1"l

ðk [ NÞ:

ð170Þ

Thus the classical inversion formula (153) applied to f 3 ðxÞ ¼ 1=½x 2 1"l takes on the form for x; l [ C, and m [ N ! " 1 ½2l "m Gðx 2 lÞ Gð1 2 lÞ : ð171Þ ¼ Dm ¼ ½x 2 1"l ½x þ m 2 1"m ½x 2 1"l Gðx þ mÞ Gð1 2 l 2 mÞ Note that (171) is a particular case of (164) for a ¼ m [ N, because, in accordance with the first formula in Remark 2, e m pi

Gðl þ mÞ Gðl þ mÞGð1 2 l 2 mÞ Gð1 2 lÞ ¼ ð21Þm GðlÞ Gð1 2 l 2 mÞ GðlÞGð1 2 lÞ ¼ cosðmpÞ

p sinðlpÞ Gð1 2 lÞ Gð1 2 lÞ ¼ : sinðl þ mÞp p Gð1 2 l 2 mÞ Gð1 2 l 2 mÞ

Hence (171) is equivalent to ! " 1 Gðx 2 lÞ Gðl þ mÞ Dm ¼ ð21Þm ½x 2 1"l Gðx þ mÞ GðlÞ

ðl [ C; m [ NÞ:

ð172Þ

Our new application is the extension of the finite difference result (172) to the generalized fractional order difference Da ð1=½x 2 1"l Þ, given by

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Corollary 6.11. Let a [ C and l [ C be such that Re(a þ l) . 0, and let Re(z) . 2 1. For the function f 3 ðxÞ U 1=½x 2 1%l , x [ R, the series given by Da ð1=½x 2 1%l Þ converges for a [ C, and has the representation for x [ R Da

!

1 ½x 2 1%l

"

¼ eapi

1 Gðx 2 lÞGða þ lÞ X Gðz þ 1 2 aÞ Gða þ 1Þ sin½p ða 2 kÞ% : ð173Þ GðlÞGðx þ aÞ k¼0 Gðz þ 1 2 kÞ Gðk þ 1Þ p ða 2 kÞ

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Proof. Noting that 1 1 ¼ Gðz þ 1 2 kÞGðk þ 1Þ Gðz þ 1Þ

z k

!

ð174Þ

;

in accordance with (73) we have for a [ C the estimate # # # 1 sin½p ða 2 kÞ%## A # #Gðz þ 1 2 kÞGðk þ 1Þ p ða 2 kÞ # # jGðz þ 1Þjk ReðzÞþ2

ðA . 0Þ;

and hence the series in the right-hand side of (173) converge for Re(z) . 2 1. Now we take the weight function

vðaÞ ¼ e2api

Gðz þ 1Þ : Gðl þ aÞGðz 2 a þ 1Þ

ð175Þ

According to (158), (169), (175) and (24), gva

!

1 ½x 2 1%l

"

Gðx 2 lÞ Gðz þ 1Þ Gðx 2 lÞ ¼ ¼ GðxÞGðlÞ Gða þ 1ÞGðz 2 a þ 1Þ GðxÞGðlÞ

z

a

!

:

By Lemma 5.4, gva ð1=½x 2 1%l Þ as a function of a is of exponential type p, gav ð1=½x 2 1%l Þ [ Bpp , and thus we can apply Theorem 6.8. By (175),

vðkÞ ¼

ð21Þk Gðz þ 1Þ Gðl þ kÞ Gðz þ 1 2 kÞ

ðk [ N0 Þ:

Substituting these expressions into the right-hand side of (159) and using (160) and (172) with m ¼ k, we have ! " 1 Gðl þ aÞGðz þ 1 2 aÞ GðxÞ Da ¼ eapi ½x 2 1%l Gðz þ 1Þ Gðx þ aÞ 1 X ð21Þk Gðz þ 1Þ Gðx þ kÞ Gðx 2 lÞ Gðl þ kÞ ð21Þk £ Gð GðxÞ Gðx þ kÞ GðlÞ l þ kÞGðz þ 1 2 kÞ k¼0 ¼ eapi which establishes (173).

1 Gðx 2 lÞGða þ lÞ X Gðz þ 1 2 aÞ Gða þ 1Þ sin½p ða 2 kÞ% ; GðlÞGðx þ aÞ k¼0 Gðz þ 1 2 kÞ Gðk þ 1Þ p ða 2 kÞ

A

Stirling functions of the first kind. Fractional Calculus

719

Example 6. As another application of Theorem 6.8 let us take the exponential function f 3 ðxÞ U elx with negative l , 0. By definition of u n one has the following recurrent formulas for u n ðelx Þ. Lemma 6.12.

For n [ N and l [ C there holds

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u n ðelx Þ ¼

n X k¼0

lðxþkÞ PðkÞ ; n ðxÞe

ð176Þ

where PðkÞ n ðxÞ ðk ¼ 0; 1; . . . ; nÞ are polynomials of degree n evaluated by the recurrent relations n Pð0Þ PðnÞ n ðxÞ ¼ ð2xÞ ; n ðxÞ ¼ xðx þ 1Þ· · ·ðx þ n 2 1Þ ðn [ NÞ; ! " ðk21Þ ðkÞ ðn [ N; k ¼ 1; 2; . . . ; n 2 1Þ: PðkÞ n ðxÞ ¼ x Pn21 ðx þ 1Þ 2 Pn21 ðxÞ

ð177Þ

ð1Þ Pð0Þ 1 ðxÞ ¼ 2x; P1 ðxÞ ¼ x;

ð179Þ

ð178Þ

In particular,

Pð0Þ 2 ðxÞ

2

¼x ;

Pð1Þ 2 ðxÞ

¼ 2xð2x þ 1Þ;

Pð2Þ 2 ðxÞ

¼ xðx þ 1Þ:

By (151), (153) can readily be shown to take the form for l [ C ! m m X m lx m2j ð21Þ D ðe Þ ¼ elðxþjÞ ¼ elx ðel 2 1Þm j j¼0

ðm [ NÞ:

ð180Þ

ð181Þ

Note that (181) is a particular case of (163) for a ¼ m [ N. Corollary 6.13. Let a [ C, l , 0 and ReðzÞ . 21. For the function f 3 ðxÞ ¼ elx , x . 0, the series given by Da elx converges for a [ C, and has the representation Da elx ¼ eapi elx ð1 2 el Þa

1 X Gðz þ 1 2 aÞ Gða þ 1Þ sin½p ða 2 kÞ& : Gðz þ 1 2 kÞ Gðk þ 1Þ p ða 2 kÞ k¼0

ð182Þ

Proof. By (174), the convergence of the series in the right-hand side of (182) is proved similarly to that in Corollary 6.11. Putting

vðaÞ ¼ e2api ð1 2 el Þ2a

Gðz þ 1Þ ; Gðx þ aÞGðz þ 1 2 aÞ

taking into account its particular case for a ¼ k [ N0

vðkÞ ¼ ð21Þk ð1 2 el Þ2k

Gðz þ 1Þ Gðx þ kÞGðz þ 1 2 kÞ

ðk [ N0 Þ;

subsituting these relations into the right-hand side of (159) and using (160) and (181) with m ¼ k, we deduce the result in (182). A

720

Remark 9.

P. L. Butzer et al.

From (169), (173) and (163), (182) we have the equality for Re(z) . 2 1 1¼

1 X Gðz þ 1 2 aÞ Gða þ 1Þ sin½p ða 2 kÞ& ; Gðz þ 1 2 kÞ Gðk þ 1Þ p ða 2 kÞ k¼0

or

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1 X 1 1 sin½p ða 2 kÞ& ¼ : Gða þ 1ÞGðz þ 1 2 aÞ k¼0 Gðk þ 1ÞGðz þ 1 2 kÞ p ða 2 kÞ

By (24), the last relation coincides with the known formula (110) apart from the multiplier G(z þ 1). Acknowledgements The authors thank the referee for his careful reading of the manuscript and especially Markus Brede (Kassel) for detecting an error in a proof which could not be repaired. The three last authors are grateful to MCYT (MTM2004-0317), to ULL, and to the Belarusian Fundamental Research Fund (F06R-104) by their support given, in part, to the present investigation.

References [1] Stirling, J., 1730, Methodus Differentialis sive Tractatus de Summatione et Interpolatione Serierum Infinitarum (London: Gul. Bowyer). [2] Butzer, P.L., Hauss, M. and Schmidt, M., 1989, Factorial functions and Stirling numbers of fractional orders. Results in Mathematics, 16(1–2), 16–48. [3] Butzer, P.L. and Hauss, M., 1991, On Stirling functions of the second kind. Studies in Applied Mathematics, 84(1), 71–91. [4] Butzer, P.L. and Hauss, M., 1991, Stirling functions of first and second kind; some new applications. In: S. Baron and D. Leviatan (Eds.) Israel Mathematical Conference Proceedings 4, Approximation, Interpolation and Summability (Israel: IMCP), pp. 89–108. [5] Butzer, P.L. and Hauss, M., 1999, Application of sampling theory to combinatorical analysis, Stirling numbers, special functions and the Riemann zeta function. In: J.R Higgins and R.L. Sterns (Eds.) Sampling Theory in Fourier Analysis: Advanced Topics (Oxford: Oxford Univ. Press), pp. 1–37. [6] Riordan, J., 1980, An Introduction to Combinatorial Analysis (New York: Wiley). [7] Aigner, M., 1975, Kombinatorik I (Berlin: Springer). [8] Comtet, L., 1974, Advanced Combinatorics: The Art of Finite and Infinite Expansion (Dordrecht, The Netherlands: Reidel Publisher). [9] Abramowitz, M. and Stegun, I.A., 1965, Handbook of Mathematical Functions (New York: Dover Publications). [10] Butzer, P.L., Kilbas, A.A. and Trujillo, J.J., 2003, Stirling functions of the second kind in the setting of difference and fractional calculus. Numerical Functional Analysis and Optimization, 24(7–8), 673– 711. [11] Butzer, P.L., Splettstoesser, W. and Stens, R.L., 1988, The sampling theorem and linear prediction in signal analysis. Jahresber Deutsche Mathematical Verein, 90, 1–70. [12] Butzer, P.L., Schmeisser, G. and Stens, R.L., 2001, An introduction to sampling analysis. In: F. Marvasti (Ed.) Nonuniform Sampling; Theory and Practice (New York: Kluwer Academic/Plenum Publishers), pp. 17 –121. [13] Butzer, P.L., 2002, The sampling theorem of signal processing. Applications to combinatorial analysis and number theory; an overview. In: A.H. Siddiqi and M. Kocvara (Eds.) Trends in Industrial and Applied Mathematics (Dordrecht: Kluwer Academic Publishers), pp. 23–61. [14] Higgins, J.R., 1996, Sampling theory in Fourier and Signal analysis; Foundations (Clarendon, Oxford: Oxford Science Publications). [15] Butzer, P.L., Kilbas, A.A. and Trujillo, J.J., 2003, Generalised Stirling functions of second kind and representations of fractional order differences via derivatives. Journal of Difference Equations and Applications, 9, 503 –533.

Downloaded by [Juan J. Trujillo] at 04:36 09 July 2011

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[16] Samko, S.G., Kilbas, A.A. and Marichev, O.I., 1993, Fractional Integrals and Derivatives. Theory and Applications (Yverdon: Gordon and Breach). ¨ ber die Theorie der fraktionierten Stirlingzahlen und deren Anwendung. Diplomarbeit, Lehrstuhl [17] Hauss, M., U A fu¨r Mathematik, RWTH Aachen, 1990, 161 pp. [18] Butzer, P.L., Markett, C. and Schmidt, M., 1991, Stirling numbers, central factorial numbers, and representations of the Riemann zeta function. Results in Mathematics, 19, 257– 275. [19] Markett, C., 1994, Triple sums and the Riemann zeta function. Journal of Number Theory, 48, 113 –132. [20] Nielsen, N., 1965, Handbuch der Theorie der Gammfunktionen (New York: Chelsea Publishing Comp). [21] Jordan, C., 1950, Calculus of Finite Differences (New York: Chelsea Publishing Company). [22] Adamchick, V.S., 1997, On Stirling numbers and Euler sums. Journal of Computational and Applied Mathematics, 79(1), 119–130. [23] Mordell, L., 1958, On the evaluation of some multiple sums. Journal of the London Mathematical Society, 33, 368– 371. [24] Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G., 1953, Higher Transcendental Functions (New York: McGraw-Hill), Vol. I. [25] Butzer, P.L. and Jansche, S., 1997, A direct approach to the Mellin transform. Journal of Fourier Analysis and Applications, 3, 325–376. [26] Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G., 1955, Higher Transcendental Functions (New York: McGraw-Hill), Vol. III. [27] Djrbashian, M.M., 1968, Integral Transforms and Representations of Functions in the Complex Domain (Russian) (Moscow: Nauka). [28] Djrbashian, M.M., 1993, Harmonic analysis and boundary value problems in the complex domain. Operator Theory: Advances and Applications (Berlin: Birkha¨usen Verlag), Vol. 65. [29] Kilbas, A.A., Srivastava, H.M. and Trrujillo, J.J., 2006, Theory and applications of fractional differential equations. North-Holland Mathematics Studies (Amsterdam: Elsevier), Vol. 204. [30] Annaby, M. and Butzer, P.L., 2004, Sampling in Paley–Wiener space associated with fractional integrodifferential operators. Journal of Computational and Applied Mathematics, 171, 39–57.