Symmetric Identities for Carlitz’s Type Higher-Order (p,q)-Genocchi Polynomials

Ahyun Kim,Cheon Seoung Ryoo

doi:10.3390/sym12101670

Abstract

In this paper, we study Carlitz’s type higher-order (p,q)-Genocchi polynomials. To be specific, we define Carlitz’s type higher-order (p,q)-Genocchi polynomials and Carlitz’s type higher-order (h,p,q)-Genocchi polynomials. This paper also explores properties including distribution relation and symmetric identities. In addition, we find alternating (p,q)-power sums. We identify symmetric identities using Carlitz’s type higher-order (h,p,q)-Genocchi polynomials and alternating (p,q)-power sums.

Highlights

The purpose of this paper is to discuss Carlitz’s type higher-order ( p, q)-Genocchi polynomials.To do so, we introduce notations and precedent researches related to the subject of this paper.We utilize the following notations: N = {1, 2, 3, · · · } denotes the set of natural numbers, Z denotes the set of integers, Z0 = N ∪ {0} denotes the set of nonnegative integers, C denotes the set of complex numbers, and ∞ ∑ = m1,···,mr =0 m1 =0 mr =0We would like to review several definitions related to q-number and ( p, q)-number used in this paper
We utilize the following notations: N = {1, 2, 3, · · · } denotes the set of natural numbers, Z denotes the set of integers, Z0 = N ∪ {0} denotes the set of nonnegative integers, C denotes the set of complex numbers, and m1,···,mr =0
For r ∈ N and 0 < q < p ≤ 1, Carlitz’s type higher-order ( p, q)-Genocchi polynomials are defined as the following generating function (r )

Summary

Introduction

The purpose of this paper is to discuss Carlitz’s type higher-order ( p, q)-Genocchi polynomials. For r ∈ N and 0 < q < p ≤ 1, Carlitz’s type higher-order ( p, q)-Genocchi polynomials are defined as the following generating function. For r ∈ N and 0 < q < p ≤ 1, Carlitz’s type higher-order (h, p, q)-Genocchi polynomials are defined as the following generating function (−q)m1 +···+mr ph(m1 +···+mr ) e[m1 +···+mr + x] p,q t. We derive symmetric properties for Carlitz’s type higher-order ( p, q)-Genocchi polynomials and the multiple ( p, q)-Hurwitz–Euler eta function. We look for the symmetric property related to the alternating ( p, q)-power sums and Carlitz’s type higher-order (h, p, q)-Genocchi polynomials.

Complex Roots

Conclusions