The Properties Of The Q-Jacobi-Stirling Numbers

The concept of Jacobi-Stirling numbers was first proposed by Everitt in 2007 who studied the spectral theory of the classical two order Jacoby differential expression.Jacobi-Stirling numbers are the coefficient of the integral complex power of Jacoby's formula in Legendre's symmetry.In 2012,Mansour proposed the q-Similar Stirling numbers,and then the related researches were carried out.Because of Stirling numbers having many similar properties with Jacobi-Stirling numbers,Legendre-Stirling numbers and q-Similar Stirling numbers,they got a lot of attention.In this paper,based on the basic functions defined in the q-Similar Stirling numbers and the expression of Jacobi-Stirling numbers and Legendre-Stirling numbers,the concepts of q-Jacobi-Stirling numbers and q-Legendre-Stirling numbers were proposed,and their related properties are studied.The main contents of this paper is the following three aspects:(1)Through the basic function[x]q=1-qx/1-q,we introduce the new "sum functions" The concepts of two kinds of q-Legendre-Stirling numbers and q-Jacobi-Stirling numbers are proposed,the recursive relations are derived.(2)It presents a matrix representation method for the first kind q-Legendre-Stirling number and the first kind q-Jacobi-Stirling number.In addition,it is proved that the q-Legendre-Stirling numbers and the q-Jacobi-Stirling numbers have some combinatorial identities.(3)The relationships between the two kinds of q-Legendre-Stirling numbers,and the relationships between the two kinds of q-Jacobi-Stirling numbers are studied which enrich the research findings of similar Stirling numbers.