On the q‐Extension of Apostol‐Euler Numbers and Polynomials

Recently, Choi et al. (2008) have studied the q ‐extensions of the Apostol‐Bernoulli and the Apostol‐Euler polynomials of order n and multiple Hurwitz zeta function. In this paper, we define Apostol′s type q ‐Euler numbers E n , q , ξ and q ‐Euler polynomials E n , q , ξ ( x ). We obtain the generating functions of E n , q , ξ and E n , q , ξ ( x ), respectively. We also have the distribution relation for Apostol′s type q ‐Euler polynomials. Finally, we obtain q ‐zeta function associated with Apostol′s type q ‐Euler numbers and Hurwitz_s type q ‐zeta function associated with Apostol′s type q ‐Euler polynomials for negative integers.