Convolution identities of poly-Cauchy numbers with level $2$

TL;DR

This paper explores convolution identities of poly-Cauchy numbers with level 2, revealing simplified forms and introducing related Stirling numbers, thereby advancing the understanding of these special number sequences.

Contribution

It establishes new convolution identities for poly-Cauchy numbers with level 2 and introduces Stirling numbers of the first kind with level 2, expanding the theoretical framework.

Findings

Convolution identities for poly-Cauchy numbers with level 2 derived.

Three poly-Cauchy numbers with level 2 can be expressed in a simple form.

Introduction of Stirling numbers of the first kind with level 2.

Abstract

Poly-Cauchy numbers with level $2$ are defined by inverse sine hyperbolic functions with the inverse relation from sine hyperbolic functions. In this paper, we show several convolution identities of poly-Cauchy numbers with level $2$. In particular, that of three poly-Cauchy numbers with level $2$ can be expressed as a simple form. In the sequel, we introduce the Stirling numbers of the first kind with level $2$