On the type 2 poly-Bernoulli polynomials associated with umbral calculus

Taekyun Kim,Jin-Woo Park,Dae San Kim,Dmitry V Dolgy

doi:10.1515/math-2021-0086

Abstract

Abstract Type 2 poly-Bernoulli polynomials were introduced recently with the help of modified polyexponential functions. In this paper, we investigate several properties and identities associated with those polynomials arising from umbral calculus techniques. In particular, we express the type 2 poly-Bernoulli polynomials in terms of several special polynomials, like higher-order Cauchy polynomials, higher-order Euler polynomials, and higher-order Frobenius-Euler polynomials.

Highlights

The poly-Bernoulli polynomials, which are defined with the help of polylogarithm functions, were studied by Kaneko in [1], while the type 2 poly-Bernoulli polynomials, which are defined with the help of modified polyexponential functions, were investigated very recently in [2]
We note that the modified polyexponential functions are inverse to the polylogarithm functions
The generating function of type 2 poly-Bernoulli polynomials is obtained in this way (see (1), (3)), and we may say that it arises in a natural manner

Summary

Introduction

The poly-Bernoulli polynomials, which are defined with the help of polylogarithm functions, were studied by Kaneko in [1], while the type 2 poly-Bernoulli polynomials, which are defined with the help of modified polyexponential functions, were investigated very recently in [2]. We note that the modified polyexponential functions are inverse to the polylogarithm functions. It is very natural to replace the polylogarithms by the modified polyexponential functions in the definition of generating function of poly-Bernoulli polynomials. The generating function of type 2 poly-Bernoulli polynomials is obtained in this way (see (1), (3)), and we may say that it arises in a natural manner. Let k ≥ 1 be an integer, Eik(x) the modified polyexponential function (see (1)), and let Bn(k) be the type 2 poly-Bernoulli numbers (see (3)). For any integer k ≥ 2, the generating function of the type 2 poly-Bernoulli numbers is given by

This work is licensed under the Creative Commons

Let be the field of complex numbers and let