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Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials. (English) Zbl 1076.33006
The authors introduce a generalization of the classical Bernoulli polynomials as analogous definition of Apostol type [see T. M. Apostol, Pac. J. Math. 1, 161–167 (1951; Zbl 0043.07103)] for the so-called Apostol-Bernoulli numbers and polynomials of higher order. The generalization, the Apostol-Bernoulli polynomials \({\mathcal B}_n^{(\alpha)}(x;\lambda)\), is defined by means of the following generating function: \[ \left(\frac{z}{\lambda\,e^z-1}\right)^{\alpha}\,e^{xz} =\sum_{n=0}^\infty {\mathcal B}_n^{(\alpha)}(x;\lambda)\,\frac{z^n}{n!}\qquad \left(| z+\log \lambda| <2\pi;\,1^{\alpha}:=1\right) \] with \[ B_{n}^{(\alpha)}(x)={\mathcal B}_n^{(\alpha)}(x;1) \quad\text{and}\quad {\mathcal B}_{n}^{(\alpha)}(\lambda):={\mathcal B}_n^{(\alpha)}(0;\lambda) \] where \({\mathcal B}_{n}^{(\alpha)}(\lambda)\) denotes the so-called Apostol-Bernoulli numbers of order \(\alpha\). In a similar manner the Apostol-Euler polynomials of order \(\alpha\), a generalization of the classical Euler polynomials, is introduced. In a previous paper, the first author derived several properties and explicit representations of the Apostol-Euler polynomials of order \(\alpha\). In this paper, the authors investigate the corresponding problems for the Apostol-Bernoulli polynomials of order \(\alpha\) by following the work of the second author in an earlier article [see H. M. Srivastava, Math. Proc. Camb. Philos. Soc. 129, 77–84 (2000; Zbl 0978.11004)]. They establish their elementary properties, derive an explicit series representations for the polynomials involving the Gaussian hypergeometric function, the Hurwitz zeta function and the Riemann zeta function.

MSC:
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
11B68 Bernoulli and Euler numbers and polynomials
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