Note on q-extensions of Euler numbers and polynomials of higher order

Abstract

In 2007, Ozden et al. constructed generating functions of higher-order twisted ( h,q)-extension of Euler polynomials and numbers, by using p-adic, q-deformed fermionic integral on p. By applying their generating functions, they derived the complete sums of products of the twisted ( h,q)-extension of Euler polynomials and numbers. In this paper, we consider the new q-extension of Euler numbers and polynomials to be different which is treated by Ozden et al. From our q-Euler numbers and polynomials, we derive some interesting identities and we construct q-Euler zeta functions which interpolate the newq-Euler numbers and polynomials at a negative integer. Furthermore, we study Barnes-typeq-Euler zeta functions. Finally, we will derive the new formula for "sums of products of q-Euler numbers and polynomials" by using fermionic p-adic, q-integral on p.