Hankel Transforms Of Some Combinatorial Sequences

The research in Hankel matrices is one of the primary topics in combinatorial matrices, including the study of Hankel transforms and decomposition of Hankel matrices. The Hankel transform arises often in combinatorics,numerical analysis,algebra and other branches of mathmatics.It also arises in computer science,structural analysis and other science.There has been an amount of interest and research devoted to the Hankel trans-form in recent years.In general,the tools to calculate Hankel transforms include the LDU decomposition of Hankel matrices,continued fractions and the theory of lattice paths.In this thesis,we show how continued fractions can be used to prove the conjectures on Hankel transforms of some sequences proposed by Barry.We also give a new proof of log-convexity and log-concavity of some sequences through the calculations of Hankel determinants.The organization of this paper is as follows:1.In the first chapter,we review the background and some related definitions of Hankel transforms.2.In the second chapter,we mainly introduce several calculational methods of Hankel transforms and liner transformations preserving Hankel transforms.We also prove that the Hankel transform is invariant under the binomial transform.3.In the third chapter,we give some proofs for the open problems about Hankel trans-forms of the series reversions proposed by Barry.4.In the forth chapter,we use the method due to Gessel-Viennot-Lindstr(o|¨)m to prove log-convexity and log-concavity of some sequences.