Total Positivity Of Recursive Arrays And Pascal-Like Arrays

The study of the total positivity has rightly received considerable attention in various branches of mathematics. It not only has provided a powerful and useful concept in studying unimodal problems, but also produced combinational inequality. This thesis would show the various properties of total positivity of two special combinatorial arrays of two classes, and it is organized as follows.In the first part, we presented the basic concepts and conclusions used in the thesis and give a brief description of totally positivity, Polya frequency sequences, and Riordan arrays.The second part is devoted to show the method that how to translate the problem of total positivity of recursive matrices into tridiagonal matrices. As consequences, we would point out that many famous combinational triangles such as Pascal triangle, Catalan triangles, Bell triangle are TP.Finally, in the third part, we would show that how to use the method we mentioned in the second part to solve the strictly total positivity problem of Pascal-like arrays in algebra way. As applications, we will show that the triangle of large Schroder numbers, Delannoy triangle, triangle of unsigned Stirling numbers of the first kind is lower strictly totally positive in a unify approach.