Generalized Legendre-Stirling Numbers


The Legendre-Stirling numbers were discovered by Everitt, Littlejohn and Wellman in 2002 in a study of the spectral theory of powers of the classical second-order Legendre differential operator. In 2008, Andrews and Littlejohn gave a combinatorial interpretation of these numbers in terms of set partitions. In 2012, Mongelli noticed that both the Jacobi-Stirling and the Legendre-Stirling numbers are in fact specializations of certain elementary and complete symmetric functions and used this observation to give a combinatorial interpretation for the generalized Legendre-Stirling numbers. In this paper we provide a second combinatorial interpretation for the generalized Legendre-Stirling numbers which more directly generalizes the definition of Andrews and Littlejohn and give a combinatorial bijection between our interpretation and the Mongelli interpretation. We then utilize our interpretation to prove a number of new identities for the generalized Legendre-Stirling numbers.

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Garrett, K. and Killpatrick, K. (2014) Generalized Legendre-Stirling Numbers. Open Journal of Discrete Mathematics, 4, 109-114. doi: 10.4236/ojdm.2014.44014.

Conflicts of Interest

The authors declare no conflicts of interest.


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[3] Everitt, W.N., Littlejohn, L.L. and Wellman, R. (2002) Legendre Polynomials, Legendre-Stirling Numbers, and the Left-Definite Spectral Analysis of the Legendre Differential Expression. Journal of Computational and Applied Mathematics, 148, 213-238.
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