Jacobi Continued Fraction Expressions Of Several Polynomial Sequences

In combinatorics,we often study properties of a coefficient sequence based on their polynomials.Therefore,polynomial is a bridge between discrete mathematics and continuous analysis.We can solve some problems in discrete mathematics with the help of continuous analysis.The study of polynomial sequences is one of classical problems in combinatorics,including the Jacobi continued fraction expression,Hankel determinant and unimodality properties of polynomial sequences.By using the Jacobi continued fraction expressions,we can obtain Hankel determinants,strong q-log convexity,3-q-log convexity,?-positivity and q-Stieltjes moment property of polynomial sequences.So Jacobi continued fraction expressions are particularly important in combinatorics.In this paper,we consider two kinds of generalized polynomial sequences(Pn(q))n?0 and(Qn(q))n?0.The sequence(Pn(q))n?0 contains a number of classical polynomial sequences,such as the Eulerian polynomials,the derangement polynomials and the binomial Eulerian polynomials for the wreath product Cr(?)Sn,the binomial Eulerian polynomials of type B and the 1/k-Eulerian polynomials of type A(B).The sequence(Qn(q))n?0 includes the polynomials of involutions in Cr(?)Sn and Hermite polynomials.In this paper,using the results of Stieltjes-Rogers and orthogonal polynomials,we obtain the Jacobi continued fraction expressions of two kinds of generalized polynomial sequences respectively.As applications,we get Hankel determinants of these two kinds of generalized polynomial sequences and strong q-log-convexity of(Pn(1))n?0.The contents of this paper are as follows:In Chapter 1,we introduce the basic concepts,the developing status and main work of this paper.In Chapter 2,we define two kinds of generalized polynomial sequences(Pn(q))n?0 and(Qn(q))n?0 firstly.Next,we prove the Jacobi continued fraction expressions of these poly-nomial sequences.Finally,by using the Jacobi continued fraction expressions,we obtain the Hankel determinants of two kinds of generalized polynomial sequences and the strong q-log-convexity of sequence(Pn(q))n?0.In Chapter 3,as applications,we get the Jacobi continued fraction expressions of the Eulerian polynomials,the derangement polynomials,the binomial Eulerian polynomials and the polynomials of involutions for the wreath product Cr(?)Sn,the binomial Eulerian poly-nomials of type B,the 1/k-Eulerian polynomials of type A(B)and Hermite polynomials.Furthermore,some other properties of these polynomials,such as Hankel determinants,strong q-log-convexity,3-q-log convexity,?-positivity and q-Stieltjes moment property,are further studied by means of the Jacobi continued fraction expressions.In Chapter 4,we present a conclusion for this paper.