New approach to $\lambda$-stirling numbers

TL;DR

This paper introduces new generating functions and identities for $mbda$-Stirling numbers and their generalizations, providing deeper combinatorial and algebraic insights into these $mbda$-analogues.

Contribution

The paper derives novel generating functions and identities for $mbda$-Stirling numbers and their r-analogues, expanding understanding of their combinatorial and algebraic properties.

Findings

New generating functions related to reciprocals of generalized rising factorials.

Derived identities for $mbda$-Stirling numbers and their r-analogues.

Connections established between $mbda$-Stirling numbers and combinatorial interpretations.

Abstract

The aim of this paper is to study the $\lambda$-Stirling numbers of both kinds which are $\lambda$-analogues of Stirling numbers of both kinds. Those numbers have nice combinatorial interpretations when $\lambda$ are positive integers. If $\lambda$ =1, then the $\lambda$-Stirling numbers of both kinds reduce to the Stirling numbers of both kinds. We derive new types of generating functions of the $\lambda$-Stirling numbers of both kinds which are related to the reciprocals of the generalized rising factorials. Furthermore, some related identities are also derived from those generating functions. In addition, all the corresponding results to the $\lambda$-Stirling numbers of both kinds are obtained also for the $\lambda$-analogues of r-Stirling numbers of both kinds which are generalizations of those numbers.