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The Operator Method, Cauchy Method and Inversion Techniques for Basic Hypergeometric Series

Author: ZhangCaiHuan
Tutor: WangJun
School: Dalian University of Technology
Course: Basic mathematics
Keywords: Basic hypergeometric series Cauchy’s method Operator method Inversion technique Summation formula Transformation formula
CLC: O173
Type: PhD thesis
Year: 2008
Downloads: 105
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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.

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