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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 n (α) (x;λ), is defined by means of the following generating function:

z λe z -1 α e xz = n=0 n (α) (x;λ)z n n!|z+logλ|<2π;1 α :=1

with

B n (α) (x)= n (α) (x;1)and n (α) (λ):= n (α) (0;λ)

where n (α) (λ) denotes the so-called Apostol-Bernoulli numbers of order α. In a similar manner the Apostol-Euler polynomials of order α, 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 α. In this paper, the authors investigate the corresponding problems for the Apostol-Bernoulli polynomials of order α 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:
33C45Orthogonal polynomials and functions of hypergeometric type
11B68Bernoulli and Euler numbers and polynomials