Generalized degenerate Bernoulli numbers and polynomials arising from Gauss hypergeometric function

Taekyun Kim,Hyunseok Lee,Lee-Chae Jang,Dae San Kim,Hanyoung Kim

doi:10.1186/s13662-021-03337-5

Taekyun Kim, Hyunseok Lee + Show 3 more

Open Access

https://doi.org/10.1186/s13662-021-03337-5

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Abstract

A new family of p-Bernoulli numbers and polynomials was introduced by Rahmani (J. Number Theory 157:350–366, 2015) with the help of the Gauss hypergeometric function. Motivated by that paper and in the light of the recent interests in finding degenerate versions, we construct the generalized degenerate Bernoulli numbers and polynomials by means of the Gauss hypergeometric function. In addition, we construct the degenerate type Eulerian numbers as a degenerate version of Eulerian numbers. For the generalized degenerate Bernoulli numbers, we express them in terms of the degenerate Stirling numbers of the second kind, of the degenerate type Eulerian numbers, of the degenerate p-Stirling numbers of the second kind and of an integral on the unit interval. As to the generalized degenerate Bernoulli polynomials, we represent them in terms of the degenerate Stirling polynomials of the second kind.

Highlights

We have witnessed in recent years that many interesting arithmetic and combinatorial results were obtained in studying degenerate versions of some special polynomials and numbers, which was initiated by Carlitz when he introduced the degenerate Stirling, Bernoulli and Euler numbers in [3]
Motivated by that paper and as a degenerate version of those numbers and polynomials, in this paper we introduce the generalized degenerate Bernoulli numbers and polynomials again in terms of the Gauss hypergeometric function which reduce to the Carlitz degenerate Bernoulli numbers and polynomials for p = 0
4 Conclusion This work was motivated by Rahmani’s paper [14] in which a new family of p-Bernoulli numbers and polynomials was constructed by means of the Gauss hypergeometric function

Summary

Introduction

We have witnessed in recent years that many interesting arithmetic and combinatorial results were obtained in studying degenerate versions of some special polynomials and numbers (see [7,8,9,10,11,12,13] and the references therein), which was initiated by Carlitz when he introduced the degenerate Stirling, Bernoulli and Euler numbers in [3]. The studies have been done with various different tools such as combinatorial methods, generating functions, umbral calculus, p-adic analysis, differential equations, special functions, probability theory and analytic number theory. It should be noted that studying degenerate versions can be done for polynomials and for transcendental functions. The degenerate gamma functions were introduced as a degenerate version of ordinary gamma functions in [9].

Objectives

Conclusion