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

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)].


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