The Riordan Arrays And Hankel Transform

Hankel matrix has been widely applied in the computational mathematics and combinatorial mathematics.Let ?r_n?_n?0 is a real sequence,the Hankel transform of ?r_n?_n?0 is written as?h_n?_n?0,where ?h_n?_n?0= det(r_i+j)_i,jⁿ=0 is a ?n+1?-order Hankel determinant of ?r_n?_n?0.Riordan matrix is an infinite lower triangular matrix,it has been widely used in the combinatorial mathematics.In this paper,we give the Hankel transform of a special kind of weighted Motzkin sequences by using two kinds of Riordan matrices.This paper contains 3 Chapters.In Chapter 1,we briefly introduce the definitions of Riordan arrays and Hankel transforms and their research status in domestic and overseas.In Chapter 2,we introduce basic knowledge related to the Riordan arrays,exponential Riordan arrays and production matrices in detail.In addition,the combinatorial interpretations of two kinds of Riordan arrays are given with the lattice path.In Chapter 3,we get an addition formula.The Hankel determinants of weighted Motzkin sequences can be computed by using this addition formula.Moreover,we generalized this conclusion.In the end,many sequences' Hankel determinants are listed in the attached table.