Moment Problems Of Combinatorial Sequences

The moment problem is one of the classical topics in mathematics.It arises naturally in analysis,probability and statistics,as well as in combinatorics.Many sequences of combinato-rial interest are known to enjoy various kinds of moment properties.For example,the Fibonacci numbers form a Hamburger moment sequence,while the Catalan numbers form a Stieltjes mo-ment sequence.The aim of this thesis is to investigate Hamburger moment and Stieltjes moment properties of combinatorial sequences.The main frame of this thesis is as follows.In the first part,we study the Hamburger moment problem.It is known that many com-binatorial sequences satisfy linear recurrence relation with constant coefficients.We present a sufficient condition for the Hamburger moment property of such sequences.As applications,we show that many famous counting numbers are Hamburger moment sequences in a unified approach,such as the Fibonacci numbers,the Pell numbers and the Lucas numbers.In addition,we introduce the concept of infinitely convexity of sequences and show that Hamburger moment sequences are infinitely convex.The second part is devoted to the Stieltjes moment problem.There have been miscellaneous characterizations of Stieltjes moment sequences.We investigate such sequences from different approaches,including the positivity criterion,the total positivity of the corresponding Hankel matrix,and the continued fraction of the generating function.Firstly,we establish some connec-tions between Stieltjes and Hamburger moment sequences via the positivity criterion.Secondly,we give a strict proof that Stieltjes moment property implies the infinitely log-convexity.Thirdly,we present a simple proof of Horn about the Stieltjes moment property from their characterization of the continued fraction.Finally,in the third part,we discuss the moment problems of Catalan-like numbers.Catalan-like numbers unify many well-known counting coefficients,including the Catalan numbers,the Motzkin numbers,the central Delannoy numbers,the central binomial(trinomial)numbers,the large(small)Schroder numbers,and so on.We provide some sufficient conditions under which the Catalan-like numbers are Hamburger or Stieltjes moment sequences in some unified ap-proach.We also give several sufficient conditions for moment properties of some Catalan-like numbers by means of generating functions.