Explicit formula for generalization of poly-Bernoulli numbers and polynomials with a,b,c parameters

Abstract

Abstract. In this paper we investigate special generalized Bernoulli polynomials with a,b,c parameters that generalize classical Bernoulli numbers and polynomials. The present paper deals with some recurrence formulae for the generalization of poly-Bernoulli numbers and polynomials with a,b,c parameters. Poly-Bernoulli numbers satisfy certain recurrence relationships which are used in many computations involving poly-Bernoulli numbers. Obtaining a closed formula for generalization of poly-Bernoulli numbers with a,b,c parameters therefore seems to be a natural and important problem. By using the generalization of poly-Bernoulli polynomials with a,b,c parameters of negative index we define symmetrized generalization of polyBernoulli polynomials with a; b; c parameters of two variables and we prove duality property for them. Also by Stirling numbers of the second kind we will find a closed formula for them. Furthermore we generalize the Arakawa-Kaneko Zeta functions and by using the Laplace-Mellin integral, define generalization of Arakawa-Kaneko Zeta functions with a,b,c parameters and obtain an interpolation formula for the generalization of poly-Bernoulli numbers and polynomials with a,b,c parameters. Furthermore we present a link between this type of Zeta functions and Dirichlet series. By our interpolation formula, we will interpolate the generalization of Arakawa-Kaneko Zeta functions with a,b,c parameters.Mathematics subject classification (2010): 11B73, 11A07.

A Note on Hermite poly-Bernoulli Numbers and Polynomials of the Second Kind

“…We also investigate and analyse its applications in number theory, combinatorics and other fields of mathematics. The results derived here are a generalization of some known summation formulae earlier studied by Jolany et al [17,18], Dattoli et al [14] and Pathan et al [29,30].…”

“…Based on the definition of Hermite polynomials and poly logarithmic function, we introduced a new class of Hermite poly-Bernoulli numbers and polynomials of the second kind. By using Jolany's methods introduced in [17] and [18], we gave Hermite poly-Bernoulli numbers and polynomials of the second kind with two variable, and also we analyse its behaviors including general symmetric properties.…”

“…One can easily see that where ( ) k n B re generalized poly-Bernoulli numbers. For more information about poly-Bernoulli numbers and poly-Bernoulli polynomials, we refer to [15][16][17][18][19][20].…”

Generalizations of poly-Bernoulli and poly-Cauchy numbers

“…In [25], shifted poly-Cauchy and poly-Bernoulli numbers are defined and in [23] these numbers are further generalized with a q parameter. In [12,16], poly-Bernoulli and poly-Cauchy numbers and polynomials are considered by means of multiparameters. The objective of this paper is to give further generalizations for poly-Bernoulli and poly-Cauchy numbers.…”

A Study of Generalized Laguerre Poly-Genocchi Polynomials

“…in References [4][5][6][7], Jolany et al introduced and studied the generalized poly-Bernoulli numbers and polynomials, which appear in the following power series…”