The authors introduce a generalization of the classical Bernoulli polynomials as analogous definition of Apostol type [see T. M. Apostol, Pac. J. Math. 1, 161–167 (1951; Zbl 0043.07103)] for the so-called Apostol-Bernoulli numbers and polynomials of higher order. The generalization, the Apostol-Bernoulli polynomials , is defined by means of the following generating function:
where denotes the so-called Apostol-Bernoulli numbers of order . In a similar manner the Apostol-Euler polynomials of order , a generalization of the classical Euler polynomials, is introduced. In a previous paper, the first author derived several properties and explicit representations of the Apostol-Euler polynomials of order . In this paper, the authors investigate the corresponding problems for the Apostol-Bernoulli polynomials of order by following the work of the second author in an earlier article [see H. M. Srivastava, Math. Proc. Camb. Philos. Soc. 129, 77–84 (2000; Zbl 0978.11004)]. They establish their elementary properties, derive an explicit series representations for the polynomials involving the Gaussian hypergeometric function, the Hurwitz zeta function and the Riemann zeta function.