Ordinary and degenerate Euler numbers and polynomials

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

doi:10.1186/s13660-019-2221-5

Abstract

In this paper, we study some identities on Euler numbers and polynomials, and those on degenerate Euler numbers and polynomials which are derived from the fermionic p-adic integrals on mathbb{Z}_{p}. Specifically, we obtain a recursive formula for alternating integer power sums and representations of alternating integer power sum polynomials in terms of Euler polynomials and Stirling numbers of the second kind, as well as various properties about Euler numbers and polynomials. In addition, we deduce representations of degenerate alternating integer power sum polynomials in terms of degenerate Euler polynomials and degenerate Stirling numbers of the second kind, as well as certain properties on degenerate Euler numbers and polynomials.

Highlights

Let p be a fixed odd prime number
We studied some identities on Euler numbers and polynomials, and those on degenerate Euler numbers and polynomials, which are derived from certain fermionic p-adic integrals on Zp
We obtained a recursive formula for alternating integer power sums and representations of alternating integer power sum polynomials in terms of Euler polynomials and of Euler polynomials together with Stirling numbers of the second kind

Summary

Introduction

Let p be a fixed odd prime number. Throughout this paper, Zp, Qp, and Cp will denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of the algebraic closure of. 1, we will review some necessary results about fermionic p-adic integrals, Euler polynomials, and alternating integer power sums. 2, we will introduce the alternating integer power sum polynomials and represent them in terms of Euler polynomials and Stirling numbers of the second kind, and derive various properties about Euler numbers and polynomials. 3, we will introduce the degenerate alternating integer power sum polynomials and express them in terms of degenerate Euler polynomials and degenerate Stirling numbers of the second, and derive some properties on degenerate Euler numbers and polynomials. Theorem 2.1 Let n, p ∈ N with n ≡ 1 (mod 2). From (1.11) and Theorem 2.1, we note the following corollary. Where N ∈ N with N ≡ 1 (mod 2) and S2(l, m) is the Stirling number of the second kind.

Some identities of degenerate Euler numbers and polynomials

Conclusions