Recursive Matrices And Generalized Narayana Polynomials

In recent years,combinatorial sequences and their recursive relations,as one of the core contents of combinatorial mathematics research,have been attracted the attention of many scholars.For example,Catalan numbers,Motzkin numbers,Narayana numbers,etc.,the combinatorial polynomials of these classical sequences satisfy a large number of combinatorial identities and recursive relations.Some matrices in combinatorial mathematics,whose general elements are mostly combinatorial and satisfy some recursive relations,are called recursive matrices.As a special matrix,recursive matrix is an important research object in combinatorial mathematics.It has been widely studied in algebra,number theory,combinatorial mathematics and other fields.Many classical combination triangles belong to recursive matrices,such as Pascal triangle,Motzkin triangle,large Schroder triangle,etc.This paper selects four kinds of recursive matrices and generalized Narayana polynomials as research objects,using the weighted partial Motzkin paths to explain a class of recursive matrices.The weighted sum of second order determinants and the weighted sum of second order product sums of a class the recursive matrices are studied by using Riordan matrix,Lagrange inversion and Residue theorem.It is studied that the generalized Narayana polynomial matrix is a recursive matrix.Not only the weighted sum of the second determinant and the weighted sum of the second product are obtained,but also the alternating weighted sum of second order determinant is obtained.When the parameters are specialized,the combinatorial identities involving classical sequences are obtained.