| Hankel matrix is one of the important research subjects in the theory of nonnegative matrices,research on whose properties is also an important topic in combinatorics.It not only often arises in classical analysis,computation of matrices,operator algebra,combinatorics and other branches of mathematics,but also has many important applications in other disciplines such as computer science.Computation of Hankel determinants is an important part of the properties of Hankel matrices,but is very hard.The main methods to calculate Hankel determinants have the orthogonal polynomials method,GesselViennotLindstr¨om Theorem,LDU decomposition of matrices,the continued fraction method and the lattice path method.In this dissertation,using exponential Riordan arrays and the orthogonal polynomials method,and the method of Sluanke and Xin,we compute the Hankel determinants of several kinds of sequences defined by the exponential generating function and ordinary generating function,respectively.The details are as follows:In Chapter 1,we introduce the research background and the basic concepts.In Chapter 2,for two types of sequence{Pn(q)}n?0and{Qn(q)}n?0defined by an exponential generating function,using the theory of exponential Riordan array and the orthogonal polynomials method,we give the Jacobi continued fraction expression of the ordinary generating of{Pn(q)}n?0and{Qn(q)}n?0.Then we get the Hankel determinant of{Pn(q)}n?0and{Qn(q)}n?0.In addition,by using the continued fractions criterion for strong qlogconvexity and 3qlogconvexity of polynomials,we give the condition that the sequence{Pn(q)}n?0and{Qn(q)}n?0is strongly qlogconvex and3qlogconvex.In Chapter 3,using the method of Sluanke and Xin,we compute the Hankel determinant of a kind of sequences defined by a q functional equation of the ordinary generating functions.This generalizes known results for Hankel determinants of many combinatorial numbers.In Chapter 4,conclusion. |
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