A Note on Type 2 Degenerate q-Euler Polynomials

Taekyun Kim,Han Young Kim,Dae San Kim,Sung-Soo Pyo

doi:10.3390/math7080681

Abstract

Recently, type 2 degenerate Euler polynomials and type 2 q-Euler polynomials were studied, respectively, as degenerate versions of the type 2 Euler polynomials as well as a q-analog of the type 2 Euler polynomials. In this paper, we consider the type 2 degenerate q-Euler polynomials, which are derived from the fermionic p-adic q-integrals on Z p , and investigate some properties and identities related to these polynomials and numbers. In detail, we give for these polynomials several expressions, generating function, relations with type 2 q-Euler polynomials and the expression corresponding to the representation of alternating integer power sums in terms of Euler polynomials. One novelty about this paper is that the type 2 degenerate q-Euler polynomials arise naturally by means of the fermionic p-adic q-integrals so that it is possible to easily find some identities of symmetry for those polynomials and numbers, as were done previously.

Highlights

We would like to introduce the type 2 degenerate q-Euler polynomials and numbers by making use of the fermionic p-adic q-integrals, as a degenerate version as well as a q-analog of the type 2 Euler polynomials and derive some basic results for them.Studying degenerate versions and q-analogs of some known special polynomials and numbers are both very good ways of naturally introducing new special polynomials and numbers
In these two ways of constructing new polynomials and numbers, the Volkenborn integrals, the fermionic p-adic integrals, the bosonic p-adic q-integrals, and the fermionic p-adic q-integrals have played very important roles and they will continue to do so. For those polynomials and numbers, we derive several expressions, generating function, relations with type 2 q-Euler polynomials and the expression corresponding to the representation of alternating integer power sum in terms of Euler polynomials
By virtue of fermionic p-adic q-integrals, we introduce type 2 degenerate q-Euler polynomials. We present for these polynomials several expressions, generating function, relations with type 2 q-Euler polynomials and the expression corresponding to the representation of alternating integer power sums in terms of Euler polynomials

Summary

Introduction

We would like to introduce the type 2 degenerate q-Euler polynomials and numbers by making use of the fermionic p-adic q-integrals, as a degenerate version as well as a q-analog of the type 2 Euler polynomials and derive some basic results for them. In these two ways of constructing new polynomials and numbers, the Volkenborn integrals ( called p-adic invariant integrals), the fermionic p-adic integrals, the bosonic p-adic q-integrals, and the fermionic p-adic q-integrals have played very important roles and they will continue to do so For those polynomials and numbers, we derive several expressions, generating function, relations with type 2 q-Euler polynomials and the expression corresponding to the representation of alternating integer power sum in terms of Euler polynomials. The type 2 degenerate Euler polynomials were defined in [1] by the following: 2y+ x +1. Motivated by Equations (12) and (15), we would like to consider the type 2 degenerate q-Euler polynomials and investigate some properties for these polynomials .

Type 2 Degenerate q-Euler Polynomials

Conclusions