## 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)].

### MSC:

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

### Citations:

Zbl 0978.11004; Zbl 0043.07103; Zbl 0621.33008
Full Text:

### References:

 [1] T.M. Apostol: “On the Lerch Zeta function”, Pacific J. Math., Vol. 1, (1951), pp. 161-167. · Zbl 0043.07103 [2] T.M. Apostol: Introduction to analytic number theory, Springer-Verlag, New York/Heidelberg/Berlin, 1976. [3] L. Comtet: Advanced Combinatorics: The Art of Finite and Infinite Expansions, Reidel, Dordrecht/Boston, 1974. (Translated from the French by J.W. Nienhuys) · Zbl 0283.05001 [4] H.M. Srivastava: “Some formulae for the Bernoulli and Euler polynomials at rational arguments”, Math. Proc. Cambridge Philos. Soc., Vol. 129, (2000), pp. 77-84. http://dx.doi.org/10.1017/S0305004100004412 · Zbl 0978.11004 [5] H.M. Srivastava and Junesang Choi: Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht/Boston/London, 2001. · Zbl 1014.33001 [6] H.M. Srivastava, P.G. Todorov: “An explicit formula for the generalized Bernoulli polynomials”, J. Math. Anal. Appl., Vol. 130, (1988), pp. 509-513. http://dx.doi.org/10.1016/0022-247X(88)90326-5 [7] H.W. Gould: “Explicit formulas for Bernoulli numbers” Amer. Math. Monthly, Vol. 79, (1972), pp. 44-51. http://dx.doi.org/10.2307/2978125 · Zbl 0227.10010 [8] Qiu-Ming Luo: “The Bernoulli Polynomials Involving the Gaussian Hypergeometric Functions”, [submitted]. · Zbl 1189.11011 [9] D. Cvijovic and J. Klinowski: “New formula for The Bernoulli and Euler polynomials at rational arguments”, Proc. Amer. Math. Soc., Vol. 123, (1995), pp. 1527-1535. http://dx.doi.org/10.2307/2161144 · Zbl 0827.11012 [10] M. Abramowitz and I.A. Stegun (Eds): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 4th printing, Washington, 1965.
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