The Promotion And Nature Of CATALAN Number, FUSS-C ATALAN Number, And SCHRODER Number

In combinatoric,Catalan numbers are a kind of important counting sequences.They are named after the Belgian mathematician Eugene Charles Catalan(1814-1894).Over the years,the research results about the Catalan numbers and their properties have emerged in endlessly.This paper aims at further studying and gen-erating the Catalan numbers and their properties based on the existent conclusions.And this paper is divided into six chapters:The first chapter introduces the research background,content and significance of the research.In the second chapter,by virtue of the Cauchy integral formula,the authors establish an integral representation for the generating function of the Catalan num-bers.From this,the authors derive some properties of the Catalan numbers,in-cluding complete monotonicity,determinantal and product inequalities.Besides,the authors derive the asymptotic expansion and some properties of the sequences involving the Catalan numbers.Meanwhile,the authors introduce(?)a generalization of the Catalan numbers named Catalan-Qi numbers.The third chapter is the study of Catalan-Qi function and its properties.In this chapter,the authors find some properties of the Catalan-Qi function,including asymptotic expansions,integral representation,complete monotonicity,logarithmi-cally complete monotonicity,generating function and inequalities.In the fourth chapter,the authors introduce a unified generalization of the Catalan numbers,the fuss numbers,the fuss-Catalan numbers and the Catalan-Qi function and discover some properties of the unified generalization.In the fifth chapter,the authors research another important counting se-quences,the Schroder numbers in combinatoric,used the similar method studying Catalan numbers in the first few chapters.What's more,the authors establish explicit formulas for the Schroder numbers and several integral representations for their generating function.The last part draws a conclusion and future work.