Note on q-Extensions of Euler Numbers and Polynomials of Higher Order

In [14] Ozden-Simsek-Cangul constructed generating functions of higher-order twisted (h, q)-extension of Euler polynomials and numbers, by using p-adic q-deformed fermionic integral on Zp. By applying their generating functions, they derived the complete sums of products of the twisted (h, q)-extension of Euler polynomials and numbers, see[13, 14]. In this paper we cosider the new q-extension of Euler numbers and polynomials to be different which is treated by Ozden-Simsek-Cangul. From our q-Euler numbers and polynomials we derive some interesting identities and we construct q-Euler zeta functions which interpolate the new q-Euler numbers and polynomials at a negative integer. Furthermore we study Barnes’ type q-Euler zeta functions. Finally we will derive the new formula for ” sums products of q-Euler numbers and polynomials” by using fermionic p-adic q-integral on Zp. 2000 Mathematics Subject Classification 11B68, 11S80 Key wordsEuler numbers, Euler polynomials, Generalized Euler numbers, Generalized Euler polynomials, q-Euler numbers and Euler polynomials, q-zeta function, Multiple q-Euler polynomials