Generalized Bernoulli numbers and a formula of Lucas

TL;DR

This paper proves a forgotten formula of Lucas for generalized Bernoulli numbers using generating functions, and applies it to derive new results involving classical Bernoulli numbers and Meixner-Pollaczek polynomials.

Contribution

It introduces a new proof and form of a sum involving Bernoulli numbers, connecting it to Meixner-Pollaczek polynomials, based on Lucas's overlooked formula.

Findings

Proved Lucas's formula for generalized Bernoulli numbers.

Derived a new form of a Bernoulli sum studied by Dilcher.

Expressed the sum in terms of Meixner-Pollaczek polynomials.

Abstract

An overlooked formula of E. Lucas for the generalized Bernoulli numbers is proved using generating functions. This is then used to provide a new proof and a new form of a sum involving classical Bernoulli numbers studied by K. Dilcher. The value of this sum is then given in terms of the Meixner-Pollaczek polynomials.