Bernoulli, poly-Bernoulli, and Cauchy polynomials in terms of Stirling and r-Stirling numbers

TL;DR

This paper reviews how Bernoulli, poly-Bernoulli, and Cauchy polynomials and numbers can be expressed using Stirling and r-Stirling numbers, highlighting their combinatorial relationships.

Contribution

It provides a comprehensive overview of expressing these special polynomials and numbers through Stirling and r-Stirling numbers, clarifying their interconnections.

Findings

Representation formulas for Bernoulli and Cauchy polynomials in terms of Stirling numbers

Connections between poly-Bernoulli numbers and Stirling numbers

Insights into the combinatorial structure of these polynomials

Abstract

We review and discuss some results on the representation of Bernoulli, poly-Bernoulli numbers, and Bernoulli and Cauchy polynomials in terms of Stirling numbers of the first or second kind, or in terms of r-Stirling numbers.