On the Apostol-Bernoulli polynomials. (English) Zbl 1073.33001

Luo, Qiu-Ming

Cent. Eur. J. Math. 2, No. 4, 509-515 (2004).

In an earlier paper [Math. Proc. Camb. Philos. Soc. 129, No. 1, 77–84 (2000; Zbl 0978.11004)], the reviewer gave explicit representations of the general Hurwitz-Lerch zeta function and the Apostol-Bernoulli polynomials \({\mathcal B}_n(x;\lambda)\) [see T. M. Apostol, Pac. J. Math. 1, 161–167 (1951; Zbl 0043.07103)] in finite series of the Hurwitz (or generalized) zeta function \(\zeta(s, a)\).
The present author repeats much of what is already contained in the reviewer’s paper. He also derives some standard properties of the Apostol-Bernoulli polynomials \({\mathcal B}_n(x;\lambda)\). One of these properties extends a result of the reviewer and P. G. Todorov [J. Math. Anal. Appl. 130, No. 2, 509–513 (1988; Zbl 0621.33008)].

Reviewer: Hari M. Srivastava (Victoria)

MSC:

11B68	Bernoulli and Euler numbers and polynomials
11M35	Hurwitz and Lerch zeta functions
33C05	Classical hypergeometric functions, \({}_2F_1\)

Keywords:

Bernoulli numbers; Bernoulli polynomials; Apostol-Bernoulli numbers; Apostol-Bernoulli polynomials; Gaussian hypergeometric functions; Stirling numbers of the second kind; Hurwitz zeta function; Lerch functional equation

Citations:

Zbl 0978.11004; Zbl 0043.07103; Zbl 0621.33008

References:

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