Generalized Legendre-Stirling Numbers

Enumerative problems are one of important research branches in combinatorics, especially many mathematicians keeping the focus on Stirling numbers, which has an important role in development and improvement of the theory of enumerative problems.The Legendre-Stirling numbers were first presented in 2002 by Everitt et al as a result of a problem involving the spectral theory of the Legendre differential expression. These numbers are the coefficients in the integral Lagrangian symmetric powers of the Legendre differential expression, which share many similar properties with the classical Stirling numbers of the second kind. It is for this reason that they are called the Legendre-Stirling numbers of the second kind. And the Legendre-Stirling numbers of the first kind were presented in 2009 by Andrews and Littlejohn. Meanwhile, they gave a combinatorial interpretation of the Legendre-Stirling numbers of the second kind. Egge gave a combinatorial interpretation of the Legendre-Stirling numbers of the first kind in 2010.On that basis this paper focuses on computational formulas of the Legendre-Stirling numbers and the generalized Legendre-Stirling numbers of the first and second kinds.The main contents of this discourse are listed as follows:(1) There are some interesting mathematical laws of the Legendre-Stirling numbers. This paper shows a relational expression of the Legendre-Stirling numbers of the first and second kinds. Three computational formulas of the Legendre-Stirling numbers of the second kind are proved in this paper. And by combining the mathematical relational expression, a computational formula of the Legendre-Stirling numbers of the first kind is deduced.(2) The two new concepts are presented by the extension of the ones of the Legendre-Stirling numbers, which are called the generalized Legendre-Stirling numbers of the first kind and the generalized Legendre-Stirling numbers of the second kind. This paper gives the definitions of the generalized Legendre-Stirling numbers, and these numbers have generating functions and recurrence relations. Some properties relating to the generalized Legendre-Stirling numbers of the first and second kinds are established by utilizing the generating functions. Moreover, a mathematical relational expression of the generalized Legendre-Stirling numbers of the first and second kinds is proved.