A family of the Apostol-type polynomials was introduced and investigated recently by Luo and Srivastava (see (Appl. Math. Comput. 217:5702-5728, 2011)). In this paper, we study this polynomial family on P, the algebra of polynomials in a single variable x over all linear functional on P. By using the way of the umbral algebra, we obtain some fundamental properties of the generalized Apostol-type polynomials. We also show some special cases which include the corresponding results of Dere and Simsek etc. MSC:05A40, 11B68, 05A10, 05A15.
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