Generalized Legendre-Stirling Numbers

Affiliation(s)

^{1}
Department of Mathematics, Statistics and Computer Science, St. Olaf College, Northfield, MN, USA.

^{2}
Natural Science Division, Pepperdine University, Malibu, CA, USA.

ABSTRACT

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.

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.

Cite this paper

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

Garrett, K. and Killpatrick, K. (2014) Generalized Legendre-Stirling Numbers.

References

[1] Mansour, T. (2012) Combinatorics of Set Partitions. Discrete Mathematics and Its Applications Series, Chapman and Hall/CRC an Imprint of Taylor and Francis LLC.

[2] Stanley, R. (2012) Enumerative Combinatorics. In Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, Vol. 49.

[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.

http://dx.doi.org/10.1016/S0377-0427(02)00582-4

[4] Andrews, G.E. and Littlejohn, L.L. (2009) A Combinatorial Interpretation of the Legendre-Stirling Numbers. Proceedings of the American Mathematical Society, 137, 2581-2590.

[5] Mongelli, P. (2012) Combinatorial Interpretations of Particular Evaluations of Complete and Elementary Symmetric Functions. Electronic Journal of Combinatorics, 19, #P60.

[6] Benjamin, A. and Quinn, J. (2003) Proofs That Really Count: The Art of Combinatorial Proof. Mathematical Association of America, Providence, RI.

[1] Mansour, T. (2012) Combinatorics of Set Partitions. Discrete Mathematics and Its Applications Series, Chapman and Hall/CRC an Imprint of Taylor and Francis LLC.

[2] Stanley, R. (2012) Enumerative Combinatorics. In Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, Vol. 49.

[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.

http://dx.doi.org/10.1016/S0377-0427(02)00582-4

[4] Andrews, G.E. and Littlejohn, L.L. (2009) A Combinatorial Interpretation of the Legendre-Stirling Numbers. Proceedings of the American Mathematical Society, 137, 2581-2590.

[5] Mongelli, P. (2012) Combinatorial Interpretations of Particular Evaluations of Complete and Elementary Symmetric Functions. Electronic Journal of Combinatorics, 19, #P60.

[6] Benjamin, A. and Quinn, J. (2003) Proofs That Really Count: The Art of Combinatorial Proof. Mathematical Association of America, Providence, RI.