On a characteristic equation of well-poised Bailey chains

In this paper, we find by inverse technique two solutions of a system of linear equations which together serve as a sufficient and necessary condition for well-poised Bailey chains. Using these two solutions, we establish a new well-poised Bailey chain, two usual Bailey chains, and a well-poised extension of Bailey’s lemma. Their applications to q-series are also investigated. X. Ma was supported by Natural Science Foundation of China (No. 10771156).     

Multiple elliptic hypergeometric series. An approach from the Cauchy determinant

A multiple generalization of elliptic hypergeometric series is studied through the Cauchy determinant for the Weierstrass sigma function. In particular, a duality transformation for multiple hypergeometric series is proposed. As an application, two types of Bailey transformations for very well-poised multiple elliptic hypergeometric series are derived.     

Abel’s lemma on summation by parts and partial <Emphasis Type="Italic">q</Emphasis>-series transformations

The partial sums of basic hypergeometric series are investigated by means of the modified Abel lemma on summation by parts. Several transformation and summation formulae for well-poised, quadratic, cubic and quartic q-series are established. This work was partially supported by National Natural Science Foundation for the Youth (Grant No. 10801026)     

普通型Bailey引理和 Riordan链

本文主要研究在超几何函数和Ramanujan-Rogers类型恒等式有非常重要作用的.Beilay引理和Riordan链的关系,证明了普通型Bailey引理本质上是一个特殊Riordan群的Riordan链.     

Transformations of well-poised hypergeometric functions over finite fields

We define a hypergeometric function over finite fields which is an analogue of the classical generalized hypergeometric series. We prove that this function satisfies many transformation and summation formulas. Some of these results are analogous to those given by Dixon, Kummer and Whipple for the well-poised classical series. We also discuss this functionʼs relationship to other finite field analogues of the classical series, most notably those defined by Greene and Katz.     

Infinite series identities derived from the very well-poised $$\Omega $$ Ω -sum

The Ramanujan Journal - For the very well-poised $$\Omega $$ -series, a universal iteration pattern is established that yields numerous infinite series identities including several important ones...     

Transformations of well-poised hypergeometric functions over finite fields

We define a hypergeometric function over finite fields which is an analogue of the classical generalized hypergeometric series. We prove that this function satisfies many transformation and summation formulas. Some of these results are analogous to those given by Dixon, Kummer and Whipple for the well-poised classical series. We also discuss this function?s relationship to other finite field analogues of the classical series, most notably those defined by Greene and Katz.     

拉格朗日反演的q模拟与Bailey引理的注记

本文主要揭示了Gessel Ira.等给出的拉格朗日反演的q—模拟形式与An-drews G.E.的Bailey引理之间的相互转化的联系,做为例证,给出了利用这些关系得到的古典超几何级数(hypergeometric series)变换和求和公式的新证明,同时得到了模5、7、9、27四个新的Roger’s-Ramanujan类型的恒等式,其具有十分重要的组合意义。     

Shifted versions of the Bailey and Well-Poised Bailey lemmas

The Bailey lemma is a famous tool to prove Rogers–Ramanujan type identities. We use shifted versions of the Bailey lemma to derive m-versions of multisum Rogers–Ramanujan type identities. We also apply this method to the Well-Poised Bailey lemma and obtain a new extension of the Rogers–Ramanujan identities.     

Bilateral inversions and terminating basic hypergeometric series identities

The q-analogue of Legendre inversions is established and generalized to bilateral sequences. They are employed to investigate the dual relations of three basic formulae due to Jackson and Bailey, on balanced ₃?₂-series, well-poised ₈?₇-series and bilateral ₆ψ₆-series. Several terminating well-poised series identities are consequently derived, including the q-Dixon formulae on terminating ₃ψ₃-series and two terminating well-poised ₅ψ₅-series identities due to [F.H. Jackson, Certain q-identities, Quart. J. Math. (Oxford) 12 (1941) 167-172; W.N. Bailey, On the analogue of Dixon’s theorem for bilateral basic hypergeometric series, Quart. J. Math. (Oxford) 1 (1950) 318-320].