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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 ${ℬ}_{n}^{\left(\alpha \right)}\left(x;\lambda \right)$, is defined by means of the following generating function:

${\left(\frac{z}{\lambda \phantom{\rule{0.166667em}{0ex}}{e}^{z}-1}\right)}^{\alpha }\phantom{\rule{0.166667em}{0ex}}{e}^{xz}=\sum _{n=0}^{\infty }{ℬ}_{n}^{\left(\alpha \right)}\left(x;\lambda \right)\phantom{\rule{0.166667em}{0ex}}\frac{{z}^{n}}{n!}\phantom{\rule{2.em}{0ex}}\left(|z+log\lambda |<2\pi ;\phantom{\rule{0.166667em}{0ex}}{1}^{\alpha }:=1\right)$

with

${B}_{n}^{\left(\alpha \right)}\left(x\right)={ℬ}_{n}^{\left(\alpha \right)}\left(x;1\right)\phantom{\rule{1.em}{0ex}}\text{and}\phantom{\rule{1.em}{0ex}}{ℬ}_{n}^{\left(\alpha \right)}\left(\lambda \right):={ℬ}_{n}^{\left(\alpha \right)}\left(0;\lambda \right)$

where ${ℬ}_{n}^{\left(\alpha \right)}\left(\lambda \right)$ 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 11B68 Bernoulli and Euler numbers and polynomials