The Group of Generalized Stirling Numbers

Abstract

In this paper we provide an algebraic approach to the generalized Stirling numbers (GSN). By defining a group S that contains the GSN, we obtain a unified interpretation for important combinatorial functions like the binomials, Stirling numbers, Gaussian polynomials. In particular we show that many GSN are products of others. We provide an explanation for the fact that many GSN appear as pairs and the inverse relations fulfilled by them. By introducing arbitrary boundary conditions, we show a Chu–Vandermonde type convolution formula for GSN. Using the group S we demonstrate a solution to the problem of finding the connection constants between two sequences of polynomials with persistent roots.