A unified presentation of three families of generalized Apostol type polynomials based upon the theory of the umbral calculus and the umbral algebra

The aim of this paper is to introduce and investigate several new identities related to a unification and generalization of the three families of generalized Apostol type polynomials such as the Apostol–Bernoulli polynomials, the Apostol–Euler polynomials and the Apostol–Genocchi polynomials. The results presented here are based upon the theory of the Umbral Calculus and the Umbral Algebra. We also introduce some operators. By using a unified generating function for these Apostol type polynomials, which was constructed recently by Özden et al. (2010) [42], we derive many new properties of these polynomials. Moreover, we give relations between these polynomials and the Stirling numbers of the first and second kind.