The Operator Method, Cauchy Method And Inversion Techniques For Basic Hypergeometric Series

Basic hypergeometric series(also called q-series)plays an important and special role in combinatorial analysis,special functions and number theory,and is widely applied in statistics and physics etc.The main content of this thesis is to find and prove some summation and transformation formulae of basic hypergeometric series by the modified Cauchy method,operator method and inversion techniques,which include many well-known results as special cases,for example,q-Saalschiitz summation formula,Bailey's ₆ψ₆ summation formula,non-terminating Watson transformation formula and Rogers-Ramanujan identities etc.The thesis consists of four chapters.In Chapter 1,we first look back the history of hypergeometric series and basic hy-pergeometric series,and then introduce some necessary concepts and notations.Chapter 2 contributes the Cauchy method.Via generalizing the Cauchy method we obtain a new method,called the modified Cauchy method.By means of this method we establish two bilateral ₃ψ₃ and ₄ψ₄ series summation formulae,two four-term summation and transformation formulae for unilateral ₃φ₂-series and bilateral ₃ψ₃-series,and two five-term summation and transformation formulae for unilateral ₃φ₂-series and bilateral ₃φ₃-series,which contain many known results as their special cases,such as non-terminating q-Saalschütz summation formula,Bilateral ₆ψ₆ series summation formula of Bailey,non-terminating Watson transformation formula and some transformations of ₃φ₂-series etc.Chapter 3 contributes the operator method.By using this method we obtain two gen-eralized transformation formulae of q-integral form,two identities of basic hypergeometric series,as well as formal extensions for q-Pfaff-Saalschütz formula,q-Chu-Vandermonde identity and a three-term transformation formula of ₃φ₂-series.In chapter 4,by the inversion technique and the series rearrangement,we find two kinds of transformation formulae of the basic hypergeometric series.One of them contains a special case of q-Dougall summation formula,the other includes the famous Rogers-Ramanujan identities as special cases.