A combinatorial interpretation of the Legendre-Stirling numbers

Authors:
George E. Andrews and Lance L. Littlejohn

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

Published electronically:
February 17, 2009

MathSciNet review:
2497469

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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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Additional Information

**George E. Andrews**

Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16801

Email:
andrews@math.psu.edu

**Lance L. Littlejohn**

Affiliation:
Department of Mathematics, Baylor University, One Bear Place #97328, Waco, Texas 76798-7328

Email:
Lance_Littlejohn@baylor.edu

DOI:
https://doi.org/10.1090/S0002-9939-09-09814-1

Keywords:
Legendre-Stirling numbers,
Stirling numbers of the second kind,
Legendre polynomials,
left-definite theory,
self-adjoint operator

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

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.