Some series identities involving the generalized Apostol type and related polynomials

A unification (and generalization) of various Apostol type polynomials was introduced and investigated recently by Luo and Srivastava [Q.-M. Luo, H.M. Srivastava, Some generalizations of the Apostol–Genocchi polynomials and the Stirling numbers of the second kind, Appl. Math. Comput. 217 (2011) 5702–5728]. In this paper, we prove several symmetry identities for these generalized Apostol type polynomials by using their generating functions. As special cases and consequences of our results, we obtain the corresponding symmetry identities for the Apostol–Euler polynomials of higher order, the Apostol–Bernoulli polynomials of higher order and the Apostol–Genocchi polynomials of higher order, and also for another family of generalized Apostol type polynomials which were investigated systematically by Ozden et al. [H. Ozden, Y. Simsek, H.M. Srivastava, A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials, Comput. Math. Appl. 60 (2010) 2779–2787]. We also derive several relations between the Apostol type polynomials, the generalized sum of integer powers and the generalized alternating sum. It is shown how each of these results would extend the corresponding known identities.