The Jacobi-Stirling Numbers

TL;DR

This paper explores the properties and combinatorial interpretations of Jacobi-Stirling numbers, linking them to spectral theory and extending classical Stirling number properties in the context of Jacobi differential expressions.

Contribution

It establishes new properties and combinatorial interpretations of Jacobi-Stirling numbers, expanding the understanding of their algebraic and spectral significance.

Findings

Jacobi-Stirling numbers share properties with classical Stirling numbers

Extended combinatorial interpretations of Jacobi-Stirling numbers

Linked Jacobi-Stirling numbers to spectral theory of differential expressions

Abstract

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.