The Linear Algebra of (r, β)-Stirling Matrices

Abstract

This paper establishes the linear algebra of the $(r, \beta)$-Stirling matrix. Along the way, this paper derives various identities, such as its factorization and relationship to the Pascal matrix and the Stirling matrix of the second kind. Additionally, this paper develops a natural extension of the Vandermonde matrix, which can be used to study and evaluate successive power sums of arithmetic progressions.