Total Positivity Of A Class Of Recursive Matrices

The total positivity of matrices is an important source of combinatorial inequalities while recursive matrices often appear in combinatorial inequalities and have many good properties.Therefore,it is of great significance to study the total positivity of recursive matrices.In this paper,we mainly study the total positivity and related analytic properties of two kinds of recursive matrices.The thesis is organized as follows.The first chapter introduces the concepts involved,the development and research status of the theory of totally positive matrices,and gives an overview of this paper.In the second chapter,we study the total positivity of two kinds of recursive matrices from the perspective of recursive relations.The first kind is the matrices satisfying a fourterm recursive relation,according to which,a sufficient condition for the total positivity of such matrices is given by induction.The second kind satisfies a five-term recursive relation.The total positivity of such matrices is first transformed into that of recursive relation matrices by the operators preserving total positivity,then to that of Toeplitz matrices and Jacobi matrices through the method of matrix decomposition.A sufficient condition is provided.With these conditions,the total positivity of many combinatorial triangles is obtained,such as Pascal triangles,Delannoy triangles,Lucas triangles,Fibonacci triangles,Jacobsthal triangles,Jacobsthal-Lucas triangles and so on.In the third chapter,the analytic properties of Jacobsthal-like triangles are disscussed.Firstly,based on the total positivity of Jacobsthal triangles and the relationship between them,we proves that Jacobsthal-like triangles are(p,q)-TP.Secondly,by using the total positivity of Riordan arrays,we prove the total positivity of the matrices extracted from the Jacobsthal triangles and the Jacobsthal-Lucas triangles.Finally,the explicit expressions of zeroes of row generating functions of Jacobsthal triangles and Jacobsthal-Lucas triangles are given by their Binet forms.According to the expressions,the asymptotic normalities of the two matrices are proved.