Identities of Symmetry for q-Euler Polynomials

Dae San Kim

doi:10.4236/ojdm.2011.11003

Abstract

In this paper, we derive eight basic identities of symmetry in three variables related to q-Euler polynomials and the q -analogue of alternating power sums. These and most of their corollaries are new, since there have been results only about identities of symmetry in two variables. These abundance of symmetries shed new light even on the existing identities so as to yield some further interesting ones. The derivations of identities are based on the p-adic integral expression of the generating function for the q -Euler polynomials and the quotient of integrals that can be expressed as the exponential generating function for the q -analogue of alternating power sums.

Highlights

Introduction andPreliminaries p f z d 1 z = pN 1 lim f N j=0 j 1 j it is easy to see thatLet p be the normalized absolute value of p, such that p p = 1 p, and let E = t p t p < 1 p 1
We derive eight basic identities of symmetry in three variables related to q-Euler polynomials and the q -analogue of alternating power sums
The derivations of identities are based on the p-adic integral expression of the generating function for the q -Euler polynomials and the quotient of integrals that can be expressed as the exponential generating function for the q -analogue of alternating power sums

Summary

Introduction and Preliminaries

We will produce 8 basic identities of symmetry in three variables w1, w2, w3 related to q-Euler polynomials and the q-analogue of alternating power sums (cf (44), (45), (48), (51), (55), (57), (59), (60)) These and most of their corollaries seem to be new, since there have been results only about identities of symmetry in two variables in the literature. The derivations of identities are based on the p-adic integral expression of the generating function for the q-Euler polynomials in (4) and the quotient of integrals in (7) that can be expressed as the exponential generating function for the q -analogue of alternating power sums We indebted this idea to the papers [5,6]

Several Types of Quotients of Fermionic Integrals

Identities for q-Euler Polynomials

Main Theorems