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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.

##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type 11B68 Bernoulli and Euler numbers and polynomials
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##### References:
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