The Properties Of The (Extended) Legendre-Stirling Numbers

The Legendre-Stirling numbers of the second kind were proposed in 2002 by Everitt et al. They are the coefficients of integral composite powers of Legendre expression in Lagrangian symmetric form, which share many similar properties with the classical Stirling numbers of the second kind. And the Legendre-Stirling numbers of the first kind were discovered in 2009 by Andrews and Littlejohn, which corresponds to the Legendre-Stirling numbers of the second kind. After that, many theories have been brought forward by many scholars. On that basis this paper focuses on computational formulas of the Legendre-Stirling numbers and the concept of the Legendre-Stirling numbers of the first kind is promoted. Then a new class of combinatorial number-the extended Legendre-Stirling numbers of the first kind is proposed.The main contents of this discourse are listed as follows:(1) This paper gives a matrix representation and the unimodality of the Legendre-Stirling numbers of the first kind and the recursive relation is proof by the operator method. And then the correlation properties of the two kinds of the Legendre-Stirling numbers are studied and a mathematical relationship between them is given.(2) The unimodality of the two kinds of the generalized Legendre-Stirling numbers and the congruence properties of two kinds of the Legendre-Stirling numbers are proved.(3) The extended Legendre-Stirling numbers of the first kind are defined via the Laurent series expansion of the function (x)-n=(x(x-2)(x-6)…(x-(n-1))-1 and then expanding the domain of the Legendre-Stirling numbers of the first kind. It can be shown that the extened Legendre-Stirling numbers have some properties similar to the Legendre-Stirling numbers of the first kind, such as the recurrence relations and high order difference properties. The relationship between the extended Legendre-Stirling numbers of the first and second kinds is obtained. Furthermore, this paper enriches the research results of the Legendre-Stirling numbers.