Identities Involving Two Kinds of q-Euler Polynomials and Numbers

We introduce two kinds of q-Euler polynomials and numbers, and investigate many of their interesting properties. In particular, we establish q-symmetry properties of these q-Euler polynomials, from which we recover the so-called Kaneko-Momiyama identity for the ordinary Euler polynomials, discovered recently by Wu, Sun, and Pan. Indeed, a q-symmetry and q-recurrence formulas among sum of product of these qanalogues Euler numbers and polynomials are obtained. As an application, from these q-symmetry formulas we deduce non-linear recurrence formulas for the product of the ordinary Euler numbers and polynomials.