q-Euler numbers and polynomials associated with p-adic q-integrals

The main purpose of this paper is to present a systemic study of some families of multiple q-Euler numbers and polynomials.In particular, by using the q-Volkenborn integration on Z p , we construct p-adic q-Euler numbers and polynomials of higher order.We also define new generating functions of multiple q-Euler numbers and polynomials.Furthermore, we construct Euler q-Zeta function.Over five decades ago, Calitz 2, 3 defined q-extension of Frobenius-Euler numbers and polynomials and proved properties analogous to those satisfied H n (u) and H n (u, x).Recently, Satoh 14, 15 used these properties, especially the so-called distribution relation