Recurrence relations of poly-Cauchy numbers by the $r$-Stirling transform

TL;DR

This paper derives formulas for poly-Cauchy numbers using the $r$-Stirling transform, connecting them with Stirling numbers of the first and second kinds, and explores annihilation formulas for negative indices.

Contribution

It introduces new formulas for poly-Cauchy numbers involving the $r$-Stirling transform and extends the theory to negative indices.

Findings

Formulas with Stirling numbers of the second kind for poly-Cauchy numbers.

Formulas with Stirling numbers of the first kind for classical and poly-Bernoulli numbers.

Discussion of annihilation formulas for poly-Cauchy numbers with negative indices.

Abstract

We give some formulas of poly-Cauchy numbers by the $r$-Stirling transform. In the case of the classical or poly-Bernoulli numbers, the formulas are with Stirling numbers of the first kind. In our case of the classical or poly-Cauchy numbers, the formulas are with Stirling numbers of the second kind. We also discuss annihilation formulas for poly-Cauchy number with negative indices.