Apostol-Euler Polynomials of Higher Order and Gaussian Hypergeometric Functions

The purpose of this paper is to give analogous definitions of Apostol type (see T. M. Apostol Pacific J. Math. 1 (1951), 161-167) for the so-called Apostol-Euler numbers and polynomials of higher order. We establish their elementary properties, obtain several explicit formulas involving the Gaussian hypergeometric function and the Stirling numbers of the second kind, and deduce their special cases and applications that lead to the corresponding formulas of the classical Euler numbers and polynomials of higher order.