Analytical aspects of a q, r-analogue of poly-Stirling numbers of both kinds

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.