The Nature Of Several Types Of Combined Sequences And Generator Functions

A combinational class play a role in mathematic.The counting sequence of a combinational class and special number are important parts of the theory of Special function and cover widely.Over the years,the research on combinational sequence and special number have emerged in endlessly.This paper aims at further studying and combinational sequence,special number and their properties based on the existent conclusions.And this paper is divided into four chapters:The first chapter introduces the research background,content and significance of the research.In the second chapter,by virtue of techniques in combinatorial analysis,the authors simplify two families of nonlinear ordinary differential equations in terms of the Stirling numbers of the first kind and find two new families of nonlinear differential equations in terms of the Stirling numbers of the second kind.Besides,the authors find several simple and explicit forms for a family of combinational sequence.The third chapter is the study of some generation functions of special numbers and its properties.In this chapter,the authors discover a representation for the derangement numbers and a closed expression the Fibonacci polynomial in terms of a tridiagonal determinant.From the determinantal representation,the authors recover several identities and recurrence relations of the derangement numbers and the Fibonacci numbers.The authors also find a simple and significant expression in terms of the Stirling numbers for derivative polynomials of a function with a parameter related to the higner order Apostol-Euler numbers and to higher order Frobenious-Euler numbers.Moreover,the authors also present a common solution to a sequence of nonlinear ordinary differential equations.The last part draws a conclusion and future work.