A closed formula for the generating function of <i>p</i>-Bernoulli numbers

Abstract

In this paper, using geometric polynomials, we obtain a generating function of p-Bernoulli numbers in terms of harmonic numbers. As consequences of this generating function, we derive closed formulas for the nite summation of Bernoulli and harmonic numbers involving Stirling numbers of the second kind. We also give a relationship between the p-Bernoulli numbers and the generalized Bernoulli polynomials.

Mathematics Subject Classication (2010): 11B68, 11B83.

Key words: Bernoulli polynomials and Bernoulli numbers, generating functions, geometric polynomials, harmonic numbers, Stirling numbers of the rst and second kind.