On Generalized Arakawa – Kaneko Zeta Functions with Parameters

For k ∈ Z, the generalized Arakawa–Kaneko zeta functions with a, b, c parameters are given by the Laplace-Mellin integral ξk(s, x; a, b, c) � (1/Γ(s)) 􏽒 ∞ 0 (Lik(1 − (ab) − )/bt − a )c t 1dt, whereR(s)> 0 and x> 0 if k≥ 1, and R(s)> 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 ξk(s, x; a, b, c) are derived.


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, Arakawa and Kaneko [1], following expression (1), defined the Arakawa-Kaneko zeta function ξ k (s) for any integers k ≥ 1 by where Li k denotes the k-th polylogarithm Li k (z) � ∞ n�1 (z n /n k ). e integral converges for R(s) > 0 and the function ξ k 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 B (k) n called poly-Bernoulli numbers. ese numbers were introduced by Kaneko [2] using the generating function: whose values, when k � 1, are just the classical Bernoulli numbers. Arakawa and Kaneko [1] showed that for m � 0, 1, 2, . . ., In [3], Coppo and Candelpergher introduced the more general Arakawa-Kaneko zeta function defined for R(s) > 0 and x > 0 by which is a very natural extension of the classical Arakawa-Kaneko zeta function. In particular, for x � 1, ξ k (s, 1) � ξ k (s) and for k � 1, ξ 1 (s, x) � sξ(s + 1, x). Note that, ξ k (s, x) satisfies the explicit formula, 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 integers n and x > 0, where poly-Bernoulli polynomials B (k) n (x) are defined by the generating function: whose values at x � 0 are precisely the poly-Bernoulli numbers. e polynomials B (k) n (x) which were introduced by Bayad and Hamahata in [4] satisfy the recurrence relation: for all k ≥ 1 and n ≥ 0.

Generalization of Arakawa-Kaneko Zeta
Functions and Poly-Bernoulli with a, b Parameters e generalization of Arakawa-Kaneko zeta functions using a, b parameters was introduced by Jolany and Corcino [5] which are defined via Laplace-Mellin integral: It is defined for R(s) > 0 and x > 0 if k ≥ 1, and for R(s) > 0 and x > |k| + 1 if k ≤ 0. Its explicit formula is given by 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: t n n! , |t| < 2π |ln a + ln b| , 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

Generalization of Poly-Bernoulli Polynomials with a, b, c Parameters
For a, b, c > 0 and a ≠ b, Jolany et al. [6] defined the generalized polynomials B (k) n (x; a, b, c) with a, b, c parameters as follows: In this section, some fundamental properties of the generalized poly-Bernoulli polynomials with a, b, c parameters are presented without proofs . ese properties are parallel to the results in [5] for the poly-Bernoulli polynomials with a, b parameters. ese results are given in the following theorems.
Theorem 1 (explicit formula). For k ∈ Z and n ≥ 0, (15) Theorem 2 (recurrence formula). For all k ≥ 1 and n ≥ 0, As a direct result, by setting a � e, b � 1, and c � e in eorem 2, the recurrence formula of the classical poly-Bernoulli polynomials (1) The next theorem presents the second recurrence formula of the generalized poly-Bernoulli polynomials with a, b, c parameters. (2)). For k ∈ Z and n ≥ 2,

Theorem 4 (recurrence formula
As a direct result, by setting a � e, b � 1, and c � e in eorem 4, the second recurrence formula of the classical poly-Bernoulli polynomials given in (1) is obtained.

Generalization of Arakawa-Kaneko Zeta Functions with a, b, c Parameters
In this section, we give the definition of the generalized Arakawa-Kaneko zeta function with a, 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 ξ k (s, x; a, b, c) will be derived.

Definition 1.
For k ∈ Z, the generalized Arakawa-Kaneko zeta functions with a, b, c parameters are given by the Laplace-Mellin integral, where R(s) > 0 and x > 0 if k ≥ 1, and for R(s) > 0 and It can be seen that ξ k (s, x; e, 1, e) is just the classical Arakawa-Kaneko zeta functions ξ k (x, s).
e following lemma gives a relation between the generalized Arakawa-Kaneko zeta functions with a, b, c parameters and the classical Arakawa-Kaneko zeta functions.
Proof. By applying Definition 1, we get By changing variables z � (ln a + ln b)t, we obtain □ Theorem 5 (interpolation formula). e function s ⟶ ξ k (s, x; a, b, c) has analytic continuation to an entire function on the whole complex s− plane and for any positive integer n, Proof. To prove that s ⟶ ξ k (s, x; a, b, c) has analytic continuation to an entire function on the whole complex s− plane, it is sufficient to show that s ⟶ ξ k (s, x) has such a property. e 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, eorem 3, and equation (7), we obtain International Journal of Mathematics and Mathematical Sciences 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 with a, b, c parameters is obtained.
Proof. Definition 1 and definition of polylogarithmic function give Consequently, As a direct result, setting a � e, b � 1, and c � e, the explicit formula of the generalized Arakawa-Kaneko zeta functions ξ k (s, x; a, b, c) reduces to the explicit formula of the classical Arakawa-Kaneko zeta functions ξ k (s, 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. is 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 with a, b, c parameters and Dirichlet series is presented. Proof e Raabe's formula for generalization of Arakawa-Kaneko zeta functions with a, b, c parameters is presented by the following theorem.
International Journal of Mathematics and Mathematical Sciences 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 with a, b, c parameters is obtained and stated as follows. □ Corollary 2. Raabe's formula in terms of generalization of poly-Bernoulli polynomials with parameters a, b, c is given as follows: Proof. By eorems 5 and 6, Hence, the desired result is obtained.

Conclusion and Recommendation
A significant result of the paper is the derivation of an interpolation formula between the generalized poly-Bernoulli polynomials with parameters a, 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 (k 1 , k 2 , . . . , k r ), where (k 1 , k 2 , . . . , k r ) 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].

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.