Degenerate polyexponential functions and type 2 degenerate poly-Bernoulli numbers and polynomials

Taekyun Kim,Jongkyum Kwon,Hyunseok Lee,Dae San Kim

doi:10.1186/s13662-020-02636-7

Abstract

The polyexponential functions were introduced by Hardy and rediscovered by Kim, as inverses to the polylogarithm functions. Recently, the type 2 poly-Bernoulli numbers and polynomials were defined by means of the polyexponential functions. In this paper, we introduce the degenerate polyexponential functions and the degenerate type 2 poly-Bernoulli numbers and polynomials, as degenerate versions of such functions and numbers and polynomials. We derive several explicit expressions and some identities for those numbers and polynomials.

Highlights

For k ∈ Z, the polyexponential function is defined by ∞ xnEik(x) = (n – 1)!nk. (1) n=1By (1), we see that Ei1(x) = ex – 1
The polyexponential function was first introduced by Hardy and is given by e(x, a|s) =
In [2, 3], Carlitz considered the degenerate Bernoulli polynomials which are given by t eλ(t)

Summary

Introduction

In [10], the type 2 poly-Bernoulli polynomials are defined by 1 et – 1 Eik log(1 + t) When x = 0, B(nk) = B(nk)(0) are called type 2 poly-Bernoulli numbers. From (1) and (2), we note that B(n1)(x) = Bn(x) (n ≥ 0), where Bn(x) are ordinary Bernoulli polynomials given by et t – In [2, 3], Carlitz considered the degenerate Bernoulli polynomials which are given by t eλ(t)

Results

Conclusion