Analytic aspects of $q,r$-analogue of poly-Stirling numbers of both kinds

TL;DR

This paper explores the combinatorial and analytical properties of $q,r$-analogues of poly-Stirling numbers of both kinds, extending classical identities and generating functions to a broader family of generalized Stirling numbers.

Contribution

It introduces new identities, recursions, and explicit formulas for $q,r$-poly-Stirling numbers, generalizing existing theorems to this broader class of numbers.

Findings

Derived new combinatorial identities and recursions.

Established explicit formulas using inclusion-exclusion.

Extended classical theorems to $q,r$-analogues of Stirling numbers.

Abstract

The Stirling numbers of type $B$ of the second kind count signed set partitions. In this paper we provide new combinatorial and analytical identities regarding these numbers as well as Broder's $r$-version of these numbers. Among these identities one can find recursions, explicit formulas based on the inclusion-exclusion principle, and also exponential generating functions. These Stirling numbers can be considered as members of a wider family of triangles of numbers that are characterized using results of Comtet and Lancaster. We generalize these theorems, which present equivalent conditions for a triangle of numbers to be a triangle of generalized Stirling numbers, to the case of the $q,r$-poly Stirling numbers, which are $q$-analogues of the restricted Stirling numbers defined by Broder and having a polynomial value appearing in their defining recursion. There are two ways to do this and these ways are related by a nice identity.