Author(s)

K. C. Garrett¹, Kendra Killpatrick^2*

¹Department of Mathematics, Statistics and Computer Science, St. Olaf College, Northfield, MN, USA.
²Natural Science Division, Pepperdine University, Malibu, CA, USA.

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.

Stirling Numbers, Legendre-Stirling Numbers, Set Partitions

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