Abstract:The Legendre-Stirling numbers were discovered in 2002 as a result of a problem involving the spectral theory of powers of the classical second-order Legendre differential expression. Specifically, these numbers are the coefficients of integral composite powers of the Legendre expression in Lagrangian symmetric form. Quite remarkably, they share many similar properties with the classical Stirling numbers of the second kind which, as shown by Littlejohn and Wellman, are the coefficients of integral powers of the Laguerre differential expression. An open question regarding the Legendre-Stirling numbers has been to obtain a combinatorial interpretation of these numbers. In this paper, we provide such an interpretation.
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- George E. Andrews
- Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16801
- MR Author ID: 26060
- Email: firstname.lastname@example.org
- Lance L. Littlejohn
- Affiliation: Department of Mathematics, Baylor University, One Bear Place #97328, Waco, Texas 76798-7328
- Email: Lance_Littlejohn@baylor.edu
- Received by editor(s): September 2, 2008
- Received by editor(s) in revised form: October 21, 2008
- Published electronically: February 17, 2009
- Communicated by: Jim Haglund
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
- Journal: Proc. Amer. Math. Soc. 137 (2009), 2581-2590
- MSC (2000): Primary 05A05, 05A15, 33C45; Secondary 34B24, 34L05, 47E05
- DOI: https://doi.org/10.1090/S0002-9939-09-09814-1
- MathSciNet review: 2497469