A generating function for a class of generalized Bernoulli polynomials

Abstract

First we derive a generating function and a Fourier expansion for a class of generalized Bernoulli polynomials. Then we derive formulas that allow certain Dirichlet series to be evaluated in terms of these generalized Bernoulli polynomials.

Generalized Fractional-Order Bernoulli Functions via Riemann-Liouville Operator and Their Applications in the Evaluation of Dirichlet Series

“…In the case, = 1, then B 1 ( ) are the generalization of the Bernoulli polynomials defined in (5). For example, when B 0 ( ) = 1 for 0 ≤ < 1, the first two fractional-order Bernoulli functions are…”

“…is the usual -th Bernoulli polynomial. Balanzario and Sanchez [5] derive the following generating function for B ( ) defined in (5):…”