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A combinatorial interpretation of the Legendre-Stirling numbers
Author(s):
George
E.
Andrews;
Lance
L.
Littlejohn
Journal:
Proc. Amer. Math. Soc.
137
(2009),
2581-2590.
MSC (2000):
Primary 05A05, 05A15, 33C45;
Secondary 34B24, 34L05, 47E05
Posted:
February 17, 2009
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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.
References:
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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:
10.1090/S0002-9939-09-09814-1
PII:
S 0002-9939(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
Posted:
February 17, 2009
Communicated by:
Jim Haglund
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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