TY - JOUR

T1 - A combinatorial interpretation of the legendre-stirling numbers

AU - Andrews, George E.

AU - Littlejohn, Lance L.

PY - 2009/8

Y1 - 2009/8

N2 - The Legendre-Stirling numbers were discovered in 2002 as a result of a problem involving the spectral theory of powers of the classical secondorder Legendre differential expression. Specifically, these numbers are the coef-ficients 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.

AB - The Legendre-Stirling numbers were discovered in 2002 as a result of a problem involving the spectral theory of powers of the classical secondorder Legendre differential expression. Specifically, these numbers are the coef-ficients 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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U2 - 10.1090/S0002-9939-09-09814-1

DO - 10.1090/S0002-9939-09-09814-1

M3 - Article

AN - SCOPUS:77951058632

VL - 137

SP - 2581

EP - 2590

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 8

ER -