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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 {\it 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 ${\Cal B}_n^{(\alpha)}(x;\lambda)$, is defined by means of the following generating function: $$ \left(\frac{z}{\lambda\,e^z-1}\right)^{\alpha}\,e^{xz} =\sum_{n=0}^\infty {\Cal B}_n^{(\alpha)}(x;\lambda)\,\frac{z^n}{n!}\qquad \left(\vert z+\log \lambda\vert <2\pi;\,1^{\alpha}:=1\right) $$ with $$ B_{n}^{(\alpha)}(x)={\Cal B}_n^{(\alpha)}(x;1) \quad\text{and}\quad {\Cal B}_{n}^{(\alpha)}(\lambda):={\Cal B}_n^{(\alpha)}(0;\lambda) $$ where ${\Cal B}_{n}^{(\alpha)}(\lambda)$ 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 {\it 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.

33C45Orthogonal polynomials and functions of hypergeometric type
11B68Bernoulli and Euler numbers and polynomials
Full Text: DOI
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