Studies On Some Problems Of Combinatorial Inversions And Q-series Transformations

The central theme of this thesis is to investigate some problems of combinatorial inversions and those of transformations of basic hypergeometric series,such as expan-sion formulas,and partial theta function identities.In order to solve these problems,we put forward some useful methods such as the Bailey operator and the substitution of parameters operator.This thesis is organized as follows.In Chapter 1,we first introduce the essential relation between the Lagrange in-version formula and combinatorial inversions,try to make a good survey on known combinatorial inversions,according as they are linear or nonlinear.At the same time,a brief introduction to the theory of the ring of formal power series and basic hyper-geometric series,both of which will be used in the subsequent chapters,are presented.Chapter 2 are some progress on combinatorial inversions made recently.First,we show that the(?,?)-inversion proposed by L.C.Hsu is sufficient but not necessary,and we also discuss its difference with the(f,g)-inversion.Second,by use of the Lagrange inversion formula,we show a nonlinear inversion formula with the Bell polynomials involved.It is a common generalization of some known results.In the rest part of this chapter we study the problem of matrix inversions which are restricted by certain conditions and set up a multi-dissected form of the(f,g)-inversion via the use of the characteristic function.Chapter 3 deals with two q-expansion formulas.One is the q-expansion formula corresponding to the special case xn?aqn and bn=bqn of the(1-xy,y-x)-expa,nsion formula,another is the expansion formula of any formal power series with respect to the base Their applications to summation and transformation formulas of basic hypergeometric series are explored in details.One of the most important results is a general symmetry transformation for well-poised Bailey pairs,from which new generating functions of the Askey-Wilson polynomials follow in a natural way.Chapter 4 is concerned with the new method of Bailey operator,which arises from a lot of existing results about Bailey pairs and Bailey lemma in the theory of basic hy-pergeometric series.As applications,some important summation and transformation formulas are establishedIn Chapter 5,we put forward the operator method of the substitution of parame-ters in order to study the underlying transformation formulas of basic hypergeometric series.The key ingredient of this method is to treat all parameters as variables.In such viewpoint,we introduce a concept of the substitution of parameters operator and investigate the allied basic properties,finally sketch a comparatively systematic frame-work for our method.As applications of this method in practice,we construct some specific substitution of parameter operators and then apply them to transformations of hypergeometric 2?1 and 302 series,thereby obtaining many original resultsChapter 6 is devoted to a special kind of infinite series in q-series-the partial theta function.We focus our attention to a few partial theta function identities due to G.E.Andrews and S.O.Warnaar,et al.,and establish a binary representation of the partial theta function,by which all these identities can be proved in a unified way.At the end of this chapter,we further extend the definition of the partial theta function to the case of two variables and investigate its basic properties.As main results,we find not only a generalization of Andrews and Warnaar's product identity but also a ternary representation for our new bivariate partial theta function.