# New results on p-Bernoulli numbers

## TL;DR

This paper explores the relationship between p-Bernoulli numbers and geometric polynomials, extending classical properties and deriving new formulas, including a Faulhaber-type summation, through integral representations.

## Contribution

It introduces new connections between p-Bernoulli numbers and geometric polynomials, extending classical Bernoulli properties and deriving novel formulas.

## Key findings

- Extended recurrence relations for p-Bernoulli polynomials
- Derived telescopic and Raabe's formulas for p-Bernoulli numbers
- Evaluated a Faulhaber-type summation using p-Bernoulli polynomials

## Abstract

We realize that geometric polynomials and p-Bernoulli polynomials and numbers are closely related with an integral representation. Therefore, using geometric polynomials, we extend some properties of Bernoulli polynomials and numbers such as recurrence relations, telescopic formula and Raabe's formula to p-Bernoulli polynomials and numbers. In particular cases of these results, we establish some new results for Bernoulli polynomials and numbers. Moreover, we evaluate a Faulhaber-type summation in terms of p-Bernoulli polynomials.