# A closed formula for the generating function of p-Bernoulli numbers

## TL;DR

This paper derives a closed-form generating function for p-Bernoulli numbers using geometric polynomials and applies it to obtain formulas for sums involving Bernoulli and harmonic numbers with Stirling numbers.

## Contribution

Introduces a new generating function for p-Bernoulli numbers and derives related closed formulas for summations involving Bernoulli and harmonic numbers.

## Key findings

- Closed-form generating function for p-Bernoulli numbers.
- Explicit formulas for finite sums of Bernoulli and harmonic numbers.
- Connections between p-Bernoulli numbers and Stirling numbers.

## Abstract

In this paper, using geometric polynomials, we obtain a generating function of p-Bernoulli numbers. As a consequences this generating function, we derive closed formulas for the finite summation of Bernoulli and harmonic numbers involving Stirling numbers of the second kind.