The values of the high order Bernoulli polynomials at integers and the r-Stirling numbers

TL;DR

This paper derives explicit formulas for high order Bernoulli numbers and polynomials at integers using r-Stirling numbers, and explores identities connecting these numbers with binomial coefficients.

Contribution

It introduces new explicit formulas and identities linking high order Bernoulli numbers, polynomials, r-Stirling numbers, and binomial coefficients.

Findings

Explicit formulas for Bernoulli numbers and polynomials at integers.

Identities connecting r-Stirling numbers and binomial coefficients.

Enhanced understanding of relationships between special number sequences.

Abstract

In this paper, we exploit the r-Stirling numbers of both kinds in order to give explicit formulae for the values of the high order Bernoulli numbers and polynomials of both kinds at integers. We give also some identities linked the r-Stirling numbers and binomial coefficients.