Q-Difference Operator And Basic Hypergeometric Series

The study of basic hypergeometric series started in 1748 when Euler considered the infinite product 1/(q;q)_∞as a generating function for p(n),the number of partitions of a positive integer n into positive integers.Later,Gauss,Heine, Rogers,Ramanujan,Watson,Slater,and many other mathematicians made great contributions in this field.Recently,thanks to Andrews and Askey's fruitful work, basic hypergeometric series becomes an active research field again.As Andrews pointed out that basic hypergeometric series has many applications to number theory,differential equation,combinatorics,statistics,physics,etc.This thesis is mainly concerned with some progress in basic hypergeometric series,including the parameter augmentation technique based on the q-difference operator and the bilateral extensions of unilateral basic hypergeometric series.In Chapter 1,we give a brief history of hypergeometric series and basic hypergeometric series,and then introduce some basic definitions and notations that will be used throughout the thesis.To make the following topics easier understandable, we display some important classical basic hypergeometric summation, transformation,as well as integral formulas.In Chapter 2,we concentrate on the parameter augmentation technique for basic hypergeometric series.The q-difference operator was introduced by Euler and was reintroduced by Rogers,Jackson,Rota,Andrews,and others to study basic hypergeometric series.In 1997 and 1998,based on the q-difference operator, Chen and Liu[42,43]constructed two q-exponential operators serve as a more efficient way,say parameter augmentation,to deal with basic hypergeometric series.In this chapter,we make a breakthrough in the parameter augmentation technique realized by introducing the Canchy operator T(a,b;D_q) and the Cauchy companion operator E(a,b;θ),which are the extensions of Chen and Liu's q-exponential operators T(bD_q) and E(bθ),respectively.By the symmetric property of some parameters in Cauchy operator identities,the transformations of Heine's _2φ1 and Sears' _3φ2 are reconstructed in a simple and natural way.In fact, employing the Cauchy operator,we can recover or even generalize many summation, transformation,and integral formulas on basic hypergeometric series.Many extensions also contain some other well-known results as special cases,e.g.,our extension of the Askey-Wilson integral also contains the Ismail-Stanton-Viennot integral,as are the extensions of the Askey-Roy integral,q-analogue of Barnes' lemmas,a summation formula of Sears,a two-term summation of Andrews,and so on.We also find that the Cauchy operator is suitable for the study of the bivariate Rogers-Szeg(o|¨) polynomials,or the continuous big q-Hermite polynomials, to derive the corresponding Mehler's formula and the Rogers formula.Although some known results are rediscovered,most of the results are new.The proofs of the known results are illustrative and elegant,and form an integral part of the work.The new results are non-trivial and significant extensions of well-known results on basic hypergeometric series.As to the Cauchy companion operator,one will encounter a convergence problem while deducing its operator identities utilizing the q-Leibniz formula. However,resorting to an expansion formula of D_qⁿ,we circumvent this difficulty and thus establish a general Cauchy companion operator identity.Compared to the operator identities obtained by Fang in[57],ours do not need the terminating restriction.The Cauchy companion operator can also be used wildly on basic hypergeometric series,we name just a few for illustration.Jackson's transformation and a new formal expansion of _3φ2 series follow immediately from the different expansion forms of the operator identities.What's more,we show that the q-Chu -Vandermonde formula and Jackson's transformation can also be extended via the Cauchy companion operator.Chapter 3 is devoted to the bilateral extensions of unilateral basic hypergeometric series.Dougall's bilateral hypergeometric summation and the q-Gauss summation are a bilateral extension and a q-analogue of Gauss' ₂F₁ summation, respectively.Motivated by finding a bilateral extension of the q-Gauss summation, or a q-extension of Dougall's summation,Bailey,Slater,Gasper,and Askey derived many important results on the bilateral _2ψ2 series.In this chapter,we establish a transformation formula,which reduces the evaluation of a _2ψ2 series to a sum of two _2φ1 series,from a _3φ2 transformation via the bilateral extension method.In some sense,this identity may be considered as a companion of two such transformations given by Slater more than half a century ago.One advan- tage of our new _2ψ2 expansion is that the right hand side,which involves the two _2φ1 series,has no apparent symmetry in parameters(but the left hand side does),thus giving rise to a number of identities as consequences,including a new bilateral extension of the q-Gauss summation which represents a _2ψ2 series as a sum of a summation and an infinite product.Besides,we find that a two-term _2ψ2 summation formula due to Slater can be recovered from a unilateral summation formula of Andrews by bilateral extension and parameter augmentation. Combining the new expansion we derived and Slater's two-term summation leads to the useful theta function identity[46,Thm.1.1],which was utilized by Chu to give a new representation of(q;q)_∞¹⁰ and therefore a new proof of Ramanujan's congruence on the partition function modulo 11.Bailey's very-well-poised _6ψ6 summation is a very powerful identity,as it stands at the top of the classical hierarchy of summation formulas for bilateral series,which also has many applications in partitions,number theory,and special functions.In[40],Chen and Fu proposed a problem of looking for a semi-finite(or even finite) form which leads to Bailey's _6ψ6 summation in a direct limit.In 2007, Jouhet[82]provided a such semi-finite form,therefore partially answered the question.With the aid of the bilateral extension method,starting from Bailey's _10φ9 transformation,we obtain two finite forms which both yield Bailey's _6ψ6 summation in a direct limit,thus give an entirely affirmative answer to Chen and Fu's question.We conclude this thesis with some classical basic hypergeometric summation and transformation formulas in the Appendix.