# On Generalized Arakawa–Kaneko Zeta Functions with Parametersa,b,c

Authors: Nestor G. Acala; Edward Rowe M. Aleluya
Journal: International Journal of Mathematics and Mathematical Sciences
 (2020)

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## Abstract

Fork∈ℤ, the generalized Arakawa–Kaneko zeta functions with a, b, c parameters are given by the Laplace-Mellin integral ξks,x;a,b,c=1/Γs∫0∞Lik1−ab−t/bt−a−tc−xtts−1dt, where ℜs>0 and x>0 if k≥1, and ℜs>0 and x>k+1 if k≤0. In this paper, an interpolation formula between these generalized zeta functions and the poly-Bernoulli polynomials with a,b,c parameters is obtained. Moreover, explicit, difference, and Raabe’s formulas for ξks,x;a,b,c are derived.


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## Body

## 1. Introduction

In the second half of the 19th century, it was well known that the Riemann zeta function may be represented by the normalized Mellin transform,(1)ζs=1Γs∫0∞ts−1et−1dt,ℜs>1.Arakawa and Kaneko [1], following expression (1), defined the Arakawa–Kaneko zeta function ξks for any integers k≥1 by(2)ξks=1Γs∫0∞Lik1−e−tet−1ts−1dt,where Lik denotes the k-th polylogarithm Likz=∑n=1∞zn/nk. The integral converges for ℜs>0 and the function ξk continues analytically to an entire function of the whole s−plane. Moreover, Arakawa and Kaneko were able to express values of this function at negative values with the aid of the generalized Bernoulli numbers Bnk called poly-Bernoulli numbers. These numbers were introduced by Kaneko [2] using the generating function:(3)Lik1−e−z1−e−z=∑n=0∞Bnkznn!,whose values, when k=1, are just the classical Bernoulli numbers. Arakawa and Kaneko [1] showed that for m=0,1,2,…,(4)ξk−m=∑l=0m−1lmlBlk.In [3], Coppo and Candelpergher introduced the more general Arakawa–Kaneko zeta function defined for ℜs>0 and x>0 by(5)ξks,x=1Γs∫0∞Lik1−e−t1−e−te−xtts−1dt,which is a very natural extension of the classical Arakawa–Kaneko zeta function. In particular, for x=1, ξks,1=ξks and for k=1,ξ1s,x=sξs+1,x.Note that,ξks,x satisfies the explicit formula,(6)ξks,x=∑m=0∞1m+1k∑j=0m−1jmj1x+js.Moreover, they also expressed the special values of this function at negative integral points by means of poly-Bernoulli polynomials, that is, for all nonnegative integersn and x>0,(7)ξk−n,x=−1nBnk−x,where poly-Bernoulli polynomials Bnkx are defined by the generating function:(8)e−xzLik1−e−t1−e−z=∑n=0∞Bnkxznn!,whose values at x=0 are precisely the poly-Bernoulli numbers. The polynomials Bnkx which were introduced by Bayad and Hamahata in [4] satisfy the recurrence relation:(9)Bnkx=∑m=0n−1mnmBn−mk−1∑l=0m−1ln−l+1mlBlx,for all k≥1 and n≥0.


## 2. Generalization of Arakawa–Kaneko Zeta Functions and Poly-Bernoulli witha,b Parameters

The generalization of Arakawa–Kaneko zeta functions usinga,b parameters was introduced by Jolany and Corcino [5] which are defined via Laplace-Mellin integral:(10)ξks,x;a,b=1Γs∫0∞Lik1−ab−tbt−a−te−xtts−1dt.It is defined forℜs>0 and x>0 if k≥1, and for ℜs>0 and x>k+1 if k≤0. Its explicit formula is given by(11)ξks,x;a,b=∑n=0∞1n+1k∑j=0nnj−1jx+jlna+j+1lnb−s.Moreover, Jolany et al. [6] introduced generalizations of poly-Bernoulli numbers and poly-Bernoulli polynomials with a,b parameters, a≠b, using the following generating functions, respectively:(12)Lik1−ab−tbt−a−t=∑n=0∞Bnka,btnn!,t<2πlna+lnb,Lik1−ab−tbt−a−text=∑n=0∞Bnkx;a,btnn!,t<2πlna+lnb.Using these new generalizations, Jolany and Corcino [5] obtained the interpolation formula between the generalized Arakawa–Kaneko zeta functions and generalized poly-Bernoulli polynomials with a,b parameters given by(13)ξk−n,x;a,b=−1nBnk−x;a,b.


## 3. Generalization of Poly-Bernoulli Polynomials witha, b, c Parameters

Fora,b,c>0 and a≠b, Jolany et al. [6] defined the generalized polynomials Bnkx;a,b,c with a,b,c parameters as follows:(14)Lik1−ab−tbt−a−tcxt=∑n=0∞Bnkx;a,b,ctnn!,t<2πlna+lnb.In this section, some fundamental properties of the generalized poly-Bernoulli polynomials witha,b,c parameters are presented without proofs . These properties are parallel to the results in [5] for the poly-Bernoulli polynomials with a,b parameters. These results are given in the following theorems.Theorem 1.
(explicit formula). Fork∈ℤ and n≥0,(15)Bnkx;a,b,c=∑m=0∞1m+1k∑j=0m−1jmjxlnc−jlna−j+1lnbn.Theorem 2.
(recurrence formula). For allk≥1 and n≥0,(16)Bnkx;a,b,c=lna+lnb∑m=0n−lnamnmBn−mk−1a,b∑l=0m−lna−ln−l+1mlBlx;a−1,b,c.As a direct result, by settinga=e, b=1, and c=e in Theorem 2, the recurrence formula of the classical poly-Bernoulli polynomials (1) is obtained.The next theorem shows the relationship of the generalized poly-Bernoulli polynomials witha, b, c parameters and the classical poly-Bernoulli polynomials.Theorem 3.
Fora,b,c>0 and n≥0,(17)Bnkx;a,b,c=lna+lnbnBnkxlnc−lnblna+lnb.
The next theorem presents the second recurrence formula of the generalized poly-Bernoulli polynomials witha, b, c parameters.Theorem 4.
(recurrence formula (2)). For k∈ℤ and n≥2,(18)B0kx;a,b,c=1;B1kx;a,b,c=12B1k−1x;a,b,c+xlnc−lnbB0kx;a,b,c;Bnkx;a,b,c=1n+1Bnk−1x;a,b,c+xlnc−lnblna+lnbn−1B0kx;a,b,c+xlnc−lnb∑m=1n−1lna+lnbn−m−1nmBmkx;a,b,c−∑m=1n−1lna+lnbn−mnm−1Bmkx;a,b,c.As a direct result, by settinga=e, b=1, and c=e in Theorem 4, the second recurrence formula of the classical poly-Bernoulli polynomials given in (1) is obtained.


## 4. Generalization of Arakawa–Kaneko Zeta Functions witha, b, c Parameters

In this section, we give the definition of the generalized Arakawa–Kaneko zeta function witha, b, c parameters and obtain an interpolation formula between these generalized zeta functions and the poly-Bernoulli polynomials with a,b,c parameters. Moreover, explicit, difference, and Raabe’s formulas for ξks,x;a,b,c will be derived.Definition 1.
Fork∈ℤ, the generalized Arakawa–Kaneko zeta functions with a, b, c parameters are given by the Laplace-Mellin integral,(19)ξks,x;a,b,c=1Γs∫0∞Lik1−ab−tbt−a−tc−xtts−1dt,where ℜs>0 and x>0 if k≥1, and for ℜs>0 and x>k+1 if k≤0.
It can be seen thatξks,x;e,1,e is just the classical Arakawa–Kaneko zeta functions ξkx,s.
The following lemma gives a relation between the generalized Arakawa–Kaneko zeta functions witha, b, c parameters and the classical Arakawa–Kaneko zeta functions.Lemma 1.
Fork∈ℤ,(20)ξks,x;a,b,c=lna+lnb−sξks,xlnc+lnblna+lnb.Proof.
By applying Definition1, we get(21)ξks,x;a,b,c=1Γs∫0∞Lik1−e−tlnab1−e−tlnabe−txlnc+lnbts−1dt.
By changing variablesz=lna+lnbt, we obtain(22)ξks,x;a,b,c=1lna+lnbs1Γs∫0∞Lik1−e−z1−e−ze−zxlnc+lnb/lna+lnbzs−1dz=1lna+lnbsξks,xlnc+lnblna+lnb.Theorem 5.
(interpolation formula). The functions⟶ξks,x;a,b,c has analytic continuation to an entire function on the whole complex s−plane and for any positive integer n,(23)ξk−n,x;a,b,c=−1nBnk−x;a,b,c.Proof.
To prove thats⟶ξks,x;a,b,c has analytic continuation to an entire function on the whole complex s−plane, it is sufficient to show that s⟶ξks,x has such a property. The details of this fact come from (7), and the interpolation formula which shows the relationship of the classical Arakawa–Kaneko zeta functions and classical poly-Bernoulli polynomials established by [3]. Hence, by using Lemma 1, Theorem 3, and equation (7), we obtain(24)ξk−n,x;a,b,c=lna+lnb−−nξk−n,xlnc+lnblna+lnb=lna+lnbn−1nBnk−xlnc−lnblna+lnb=−1nBnk−x;a,b,c.
As an immediate consequence of the previous theorems in this section, the explicit formula for the generalized Arakawa–Kaneko zeta functions witha, b, c parameters is obtained.Corollary 1.
(explicit formula). Fork∈ℤ,(25)ξks,x;a,b,c=∑n=0∞1n+1k∑j=0nnj−1jxlnc+jlna+j+1lnb−s.Proof.
Definition1 and definition of polylogarithmic function give(26)ξks,x;a,b,c=1Γs∫0∞Lik1−ab−tbt−a−tc−xtts−1dt=1Γs∫0∞Lik1−e−tlnab1−e−tlnabe−txlnc+lnbts−1dt=1Γs∫0∞∑n=0∞1−e−tlnabnn+1ke−txlnc+lnbts−1dt.
Consequently,(27)ξks,x;a,b,c=1Γs∑n=0∞1n+1k∫0∞1−e−tlnabne−txlnc+lnbts−1dt=1Γs∑n=0∞1n+1k∫0∞∑j=0nnj−1je−txlnc+jlna+j+1lnbts−1dt=∑n=0∞1n+1k∑j=0nnj−1j⋅1Γs∫0∞e−txlnc+jlna+j+1lnbts−1dt=∑n=0∞1n+1k∑j=0nnj−1j1xlnc+jlna+j+1lnbs.
As a direct result, settinga=e, b=1, and c=e, the explicit formula of the generalized Arakawa–Kaneko zeta functions ξks,x;a,b,c reduces to the explicit formula of the classical Arakawa–Kaneko zeta functions ξks,x which is given in equation (6).
Raabe’s formula is a fundamental and universal property in the theory of zeta functions and plays an important role in special functions. This formula holds for several types of zeta functions and provides a powerful link between zeta integrals and Dirichlet series. In the next theorem, an interesting link between integral of the generalized Arakawa–Kaneko zeta functions witha, b, c parameters and Dirichlet series is presented.Lemma 2.
(difference formula)(28)ξks,x+lnablnc;a,b,c−ξks,x;a,b,c=∑m=0∞1m+1k∑j=0m+1m+1j−1j+1xlnc+jlna+j+1lnb−s.Proof.
(29)
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The Raabe’s formula for generalization of Arakawa–Kaneko zeta functions witha, b, c parameters is presented by the following theorem.Theorem 6.
(Raabe’s formula)(30)∫0lnab/lncξks,x+w;a,b,cdw=1s−1lnc∑m=0∞1m+1k∑j=0m+1m+1j−1jxlnc+jlna+j+1lnb1−s.Proof.
(31)
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+
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;
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Hence, utilizing Lemma2 results in(32)∫0lnab/lncξks,x+w;a,b,cdw=1lnc⋅Γs−1Γs1Γs−1∫0∞Lik1−e−tlnabe−txlnc+lnbts−2dt=1s−1lncξks−1,x;a,b,c−ξks−1,x+lnablnc;a,b,c=1s−1lnc∑m=0∞1m+1k∑j=0m+1m+1j−1jxlnc+jlna+j+1lnb1−s.
As a direct result of Raabe’s formula and interpolation formula, the following corollary that presents the Raabe’s formula in terms of the generalized poly-Bernoulli polynomials witha, b, c parameters is obtained and stated as follows.Corollary 2.
Raabe’s formula in terms of generalization of poly-Bernoulli polynomials with parametersa, b, c is given as follows:(33)∫0lnab/lncBnk−x−w;a,b,cdw=−1n+1n+1lnc∑m=0∞1m+1k∑j=0m+1m+1j−1jxlnc+jlna+j+1lnbn+1.Proof.
By Theorems5 and 6,(34)∫0lnab/lncBnk−x−w;a,b,cdw=∫0lnab/lnc−1nξk−n,x+w;a,b,cdw=−1n∫0lnab/lncξk−n,x+w;a,b,cdw=−1n+1n+1lnc∑m=0∞1m+1k∑j=0m+1m+1j−1jxlnc+jlna+j+1lnbn+1.
Hence, the desired result is obtained.


## 5. Conclusion and Recommendation

A significant result of the paper is the derivation of an interpolation formula between the generalized poly-Bernoulli polynomials with parametersa,b,c and the generalized Arakawa–Kaneko zeta functions with parameters a,b,c. By applying this interpolation formula, several properties of the generalized Arakawa–Kaneko zeta functions with parameters a,b,c are established. It seems interesting to define another generalization of Arakawa–Kaneko zeta functions with parameters a,b,c through multiple polylogarithms of index set k1,k2,…,kr, where k1,k2,…,kr are any r-tuples of positive integers, and relate these to the multi-poly-Bernoulli polynomials of parameters a,b,c introduced by Corcino et al. in [7].



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