Some Formulas for Apostol-euler Polynomials Associated with Hurwitz Zeta Function at Rational Arguments

Throughout this paper, we always make use of the following notation: N = {1, 2, 3, . . .} denotes the set of natural numbers, N0 = {0, 1, 2, 3, . . .} denotes the set of nonnegative integers, Z−0 = {0,−1,−2,−3, . . .} denotes the set of nonpositive integers, Z denotes the set of integers, R denotes the set of real numbers, C denotes the set of complex numbers. The generalized Bernoulli polynomialsB (α) n (x) and Euler polynomialsE (α) n (x) of order α (real or complex) are usually defined by means of the following generating functions (see, for details, [1, 5, 13, 15]):