Some Symmetry Identities for Carlitz’s Type Degenerate Twisted (p,q)-Euler Polynomials Related to Alternating Twisted (p,q)-Sums

Cheon-Seoung Ryoo

doi:10.3390/sym13081371

Abstract

In this paper, we define a new form of Carlitz’s type degenerate twisted (p,q)-Euler numbers and polynomials by generalizing the degenerate Euler numbers and polynomials, Carlitz’s type degenerate q-Euler numbers and polynomials. Some interesting identities, explicit formulas, symmetric properties, a connection with Carlitz’s type degenerate twisted (p,q)-Euler numbers and polynomials are obtained. Finally, we investigate the zeros of the Carlitz’s type degenerate twisted (p,q)-Euler polynomials by using computer.

Highlights

Many mathematicians have been working in the fields of the degenerate Euler numbers and polynomials, degenerate Bernoulli numbers and polynomials, degenerate tangent numbers and polynomials, degenerate Genocchi numbers and polynomials, degenerate Stirling numbers, and special polynomials
We recall that the degenerate Euler numbers En(μ) and Euler polynomials En(z, μ), which are defined by generating functions like following:
In our previous paper [13], we studied some identities of symmetry on the Carlitztype twisted (p, q)-Euler numbers and polynomials

Summary

Introduction

Many mathematicians have been working in the fields of the degenerate Euler numbers and polynomials, degenerate Bernoulli numbers and polynomials, degenerate tangent numbers and polynomials, degenerate Genocchi numbers and polynomials, degenerate Stirling numbers, and special polynomials (see [1,2,3,4,5,6,7,8]). In this paper we define a new form of Carlitz’s type degenerate twisted (p, q)-Euler numbers and polynomials and study some theories of the Carlitz’s type degenerate twisted (p, q)-Euler numbers and polynomials. We recall that the degenerate Euler numbers En(μ) and Euler polynomials En(z, μ), which are defined by generating functions like following (see [2,3,4]):. (1 + +1 z μt) μ , We remind that well-known Stirling numbers of the first kind S1(n, j) and the second kind S2(n, j) are defined by this (see [2,4,6]).

We also have

By substitute az ai b for z in Definition

Degree n Real Zeros Complex Zeros Real Zeros Complex Zeros

Conclusions and Discussion