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 
Quotes  0 
Basic hypergeometric series（also called qseries）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 wellknown results as special cases,for example,qSaalschiitz summation formula,Bailey’s _{6}ψ_{6} summation formula,nonterminating Watson transformation formula and RogersRamanujan identities etc.The thesis consists of four chapters.In Chapter 1,we first look back the history of hypergeometric series and basic hypergeometric 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 _{3}ψ_{3} and _{4}ψ_{4} series summation formulae,two fourterm summation and transformation formulae for unilateral _{3}φ_{2}series and bilateral _{3}ψ_{3}series,and two fiveterm summation and transformation formulae for unilateral _{3}φ_{2}series and bilateral _{3}φ_{3}series,which contain many known results as their special cases,such as nonterminating qSaalschütz summation formula,Bilateral _{6}ψ_{6} series summation formula of Bailey,nonterminating Watson transformation formula and some transformations of _{3}φ_{2}series etc.Chapter 3 contributes the operator method.By using this method we obtain two generalized transformation formulae of qintegral form,two identities of basic hypergeometric series,as well as formal extensions for qPfaffSaalschütz formula,qChuVandermonde identity and a threeterm transformation formula of _{3}φ_{2}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 qDougall summation formula,the other includes the famous RogersRamanujan identities as special cases.