Some identities on $\lambda$-analogues of $r$-stirling numbers of the second kind

TL;DR

This paper introduces and explores the properties of $mbda$-analogues of $r$-Stirling numbers of the second kind, including recurrence relations, identities, and related polynomials, extending previous work on the first kind.

Contribution

It defines the $mbda$-analogues of $r$-Stirling numbers of the second kind and investigates their properties, recurrence relations, identities, and related Dowling polynomials.

Findings

Derived recurrence relations for the $mbda$-analogues.

Established identities and properties of these numbers.

Presented a Dobinski-like formula for the associated polynomials.

Abstract

Recently, the $\lambda$-analogues of $r$-Stirling numbers of the first kind were studied by Kim-Kim. The aim of this paper is to introduce the $\lambda$-analogues of $r$-Stirling numbers of the second kind and to investigate some properties, recurrence relations and certain identities on those numbers. We also introduce the $\lambda$-analogues of Whitney-type $r$-Stirling numbers of the second and derive similar results to the case of the $\lambda$-analogues of r-Stirling numbers of the second kind. In addition, we consider the $\lambda$-analogues of Dowling polynomials and deduce a Dobinski-like formula.