  The Jacobi-Stirling numbers were discovered as a result of a problem
involving the spectral theory of powers of the classical second-order Jacobi
differential expression. Specifically, these numbers are the coefficients of
integral composite powers of the Jacobi expression in Lagrangian symmetric
form. Quite remarkably, they share many properties with the classical Stirling
numbers of the second kind which, as shown in LW, are the coefficients of
integral powers of the Laguerre differential expression. In this paper, we
establish several properties of the Jacobi-Stirling numbers and its companions
including combinatorial interpretations thereby extending and supplementing
known contributions to the literature of Andrews-Littlejohn,
Andrews-Gawronski-Littlejohn, Egge, Gelineau-Zeng, and Mongelli.
