A note on degenerate Genocchi and poly-Genocchi numbers and polynomials

Taekyun Kim,Jongkyum Kwon,Dae San Kim,Han Young Kim

doi:10.1186/s13660-020-02378-w

Abstract

Recently, Dolgy–Jang introduced the poly-Genocchi polynomials and numbers arising from the modified polyexponential function. In this paper, we study the degenerate poly-Genocchi polynomials and numbers constructed from the modified degenerate polyexponential function. We derive explicit expressions for those polynomials and numbers. Also, we obtain identities involving those polynomials and numbers and some other special numbers and polynomials. In addition, we investigate the higher-order degenerate Genocchi polynomials and find identities involving those polynomials and the higher-order Changhee polynomials.

Highlights

In [2], Carlitz initiated a study of degenerate versions of some special polynomials and numbers, namely the degenerate Bernoulli and Euler polynomials and numbers
The aim of this paper is to introduce a degenerate version of the poly-Genocchi polynomials and numbers, so-called degenerate poly-Genocchi polynomials and numbers, constructing from the modified degenerate polyexponential function
3 Conclusion In this paper, the degenerate poly-Genocchi polynomials were introduced by means of the modified degenerate polyexponential function

Summary

Introduction

In [2], Carlitz initiated a study of degenerate versions of some special polynomials and numbers, namely the degenerate Bernoulli and Euler polynomials and numbers. We have witnessed in recent years that much research has been done for various degenerate versions of many special polynomials and numbers. These include the degenerate Stirling numbers of the first and second kinds, degenerate Bernoulli numbers of the second kind, degenerate Bell numbers and polynomials, degenerate complete Bell polynomials and numbers, degenerate central factorial numbers of the second kind, degenerate Bernstein polynomials, degenerate random variables, and so on (see [6, 7, 11,12,13,14,15, 17,18,19, 21, 22, 25, 26] and the references therein). 1, we recall some necessary stuffs that are needed throughout this paper These include the degenerate exponential functions, the degenerate Genocchi polynomials, the degenerate Euler polynomials, and the degenerate Stirling numbers of the first and second kinds.

Objectives

Conclusion