P-Adic integral on mathbb{Z}_{p} associated with degenerate Bernoulli polynomials of the second kind

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

doi:10.1186/s13662-020-02746-2

Abstract

In this paper, by means of p-adic Volkenborn integrals we introduce and study two different degenerate versions of Bernoulli polynomials of the second kind, namely partially and fully degenerate Bernoulli polynomials of the second kind, and also their higher-order versions. We derive several explicit expressions of those polynomials and various identities involving them.

Highlights

Introduction and preliminariesIn [1, 2], Carlitz studied degenerate versions of Bernoulli and Euler polynomials, namely the degenerate Bernoulli and Euler polynomials, and obtained some interesting arithmetic and combinatorial results
In [18] we demonstrated that both the degenerate Stirling polynomials of the second and the r-truncated degenerate Stirling polynomials of the second kind appear in certain expressions of the probability distributions of appropriate random variables
The second one is their possible application to differential equations from which some useful identities follow

Summary

Introduction

Introduction and preliminariesIn [1, 2], Carlitz studied degenerate versions of Bernoulli and Euler polynomials, namely the degenerate Bernoulli and Euler polynomials, and obtained some interesting arithmetic and combinatorial results. Various degenerate versions of many special polynomials and numbers regained interest of some mathematicians, and quite a few results have been discovered. These include the degenerate Stirling numbers of the first and second kinds, degenerate central factorial numbers of the second kind, degenerate Bernoulli numbers of the second kind, degenerate Bernstein polynomials, degenerate Bell numbers and polynomials, degenerate central Bell numbers and polynomials, degenerate complete Bell polynomials and numbers, degenerate Cauchy numbers, and so on (see [3, 10, 13, 16, 18, 19] and the references therein). The possible applications of our results are discussed in the last section

Objectives

Conclusion