Sums of Products of q-Euler Polynomials and Numbers

The purpose of this paper is to derive formulae for the sums of products of the q-Euler polynomials and numbers, since many identities can be obtained from our sums of products of the q-Euler polynomials and numbers. In 1 , Simsek evaluated the complete sums for the Euler numbers and polynomials and obtained some identities related to Euler numbers and polynomials from his complete sums, and Jang et al. 2 also considered the sums of products of Euler numbers. Kim 3 derived the sums of products of the q-Euler numbers using the fermionic p-adic q-Volkenborn integral. In this paper, we will evaluate the complete sum of the q-Euler polynomials and numbers using the fermionic p-adic q-Volkenborn integral on Zp. Assume that p is a fixed odd prime. Throughout this paper, the symbols Zp,Qp,C, and Cp denote the ring of p-adic rational integers, the field of p-adic rational numbers, the complex number field, and the completion of algebraic closure of Qp, respectively. Let N be the set of natural numbers. Let vp be the normalized exponential valuation of Cp with |p|p p−vp p p−1. When one talks about q-extension, q is variously considered as an indeterminate, which is a complex number q ∈ C, or a p-adic number q ∈ Cp. If q ∈ C, one normally assumes |q| < 1. If q ∈ Cp, then one assumes |q − 1|p < 1. We use the notations