| Riordan matrices are a very important class of matrices arise in combinatorial mathematics.In addition to their own good properties,they can also be used as an important tool to study other combinatorial objects.For example,Riordan matrix is used to describe combinatorial identity,inverse relation and pass-path counting,etc.An important combinatorial property of Riordan matrix,its total positivity,has been widely studied.The discriminant theorem of its total positivity is described in terms of its generating function,coefficient matrix,and A series and Z series.The total positivity of matrices is closely related to unimodal problems.Delannoy matrices are special Riordan matrices with many interesting properties.It is also completely positive,and its row sequence is PF sequence.The real root of polynomials is an important research topic in combinatorial analysis.It can characterize finite PF sequences and provide a research tool for asymptotic normality.In this paper,we mainly study the total positivity of Riordan matrices,and define the generalized Delannoy matrices on the basis of Delannoy matrices.Then we study the real roots of the row polynomials of the generalized Delannoy matrices.This paper is mainly divided into three parts as follows:The first part introduces some basic definitions and development background needed in this paper.It makes a good knowledge preparation for the following research to understand well of Riordan matrix,generalized Delannoy matrix,and related combinational properties,which are our main research objects.In the second part,the characterization of the total positivity of Bell type Riordan matrices and general Riordan matrices is given.According to the particularity of sequence A and sequence Z of Bell Riordan matrices,the generating function of sequence is decomposed,and A new method to judge its total positivity is given.On this basis,we further generalize by considering the general type of Riordan matrix with both A sequence and Z sequence here.Finally,we show its applications.In the third part,we give the case of the roots of row polynomials in generalized Delannoy matrices.We prove that the roots of row polynomials in this kind of matrices are real roots and have the properties of interleaving and denseness by using the discriminating method of the interleaving of the roots of real polynomials,and the limit judgment theorem of the zeros of polynomial sequences which are not always zero. |
