Abstract
Abstract Recently, Tremblay, Gaboury and Fugère introduced a class of the generalized Bernoulli polynomials (see Tremblay in Appl. Math. Let. 24:1888-1893, 2011). In this paper, we introduce and investigate an extension of the generalized Apostol-Euler polynomials. We state some properties for these polynomials and obtain some relationships between the polynomials and Apostol-Bernoulli polynomials, Stirling numbers of the second kind, Jacobi polynomials, Laguerre polynomials, Hermite polynomials and generalized Bernoulli polynomials. MSC:11B68, 11B73, 33C45.
Highlights
Tremblay, Gaboury and Fugère introduced a class of the generalized Bernoulli polynomials
1 Introduction, definitions and motivation The generalized Bernoulli polynomials Bn(α)(x) of order α ∈ Z and the generalized Euler polynomials En(α)(x) of order α ∈ Z are defined by the following generating functions
Definition . (Luo and Srivastava [ ]) The generalized Apostol-Bernoulli polynomials Bn(α)(x; λ) of order α ∈ N are defined by means of the following generating function: t λet
Summary
Tremblay, Gaboury and Fugère introduced a class of the generalized Bernoulli polynomials Luo and Srivastava introduced the generalized Apostol-Bernoulli polynomials Bn(α)(x; λ) and the generalized Apostol-Euler polynomials En(α)(x; λ) as follows. (Luo and Srivastava [ ]) The generalized Apostol-Bernoulli polynomials Bn(α)(x; λ) of order α ∈ N are defined by means of the following generating function: t λet –
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