Some series identities involving the generalized Apostol type and related polynomials. (English) Zbl 1236.33020
Summary: A unification (and generalization) of various Apostol type polynomials was introduced and investigated recently by Q.-M. Luo and the second author [Appl. Math. Comput. 217, No. 12, 5702–5728 (2011; Zbl 1218.11026)]. 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 H. Özden, Y. Simsek and H.M. Srivastava [Comput. Math. Appl. 60, No. 10, 2779–2787 (2010; Zbl 1207.33015)]. 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.
MSC:
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 11B83 | Special sequences and polynomials |
Keywords:
generalized Apostol type polynomials; generalized Apostol-Euler polynomials; generalized Apostol-Bernoulli polynomials; Genocchi polynomials of higher order; generalized sum of integer powers; generalized alternating sumReferences:
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