The Jacobi-Stirling numbers

Authors:
George E. Andrews;Eric S. Egge;Wolfgang Gawronski;Lance L. Littlejohn
Affiliations:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16801, United States;Department of Mathematics, Carleton College, Northfield, MN 55057, United States;Department of Mathematics, University of Trier, 54286 Trier, Germany;Department of Mathematics, Baylor University, One Bear Place #97328, Waco, TX 76798-7328, United States
Venue:
Journal of Combinatorial Theory Series A
Year:
2013

Citing 6
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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 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 recent contributions to the literature.