Vincent Y. B. Chen, William Y. C. Chen, Nancy S. S. Gu
We obtain a formula which reduces the evaluation of a $_2ψ_2$ series to two $_2ϕ_1$ series. In some sense, this identity may be considered as a companion of Slater's formulas. We also find that a two-term ${}_2ψ_2$ summation formula due to Slater can be derived from a unilateral summation formula of Andrews by bilateral extension and parameter augmentation.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We give elementary derivations of several classical and some new summation and transformation formulae for bilateral basic hypergeometric series. For purpose of motivation, we review our previous simple proof ("A simple proof of Bailey's very-well-poised 6-psi-6 summation", Proc. Amer. Math. Soc., to appear) of Bailey's very-well-poised 6-psi-6 summation. Using a similar but different method, we now give elementary derivations of some transformations for bilateral basic hypergeometric series. In particular, these include M. Jackson's very-well-poised 8-psi-8 transformation, a very-well-poised …
We prove some new semi-finite forms of bilateral basic hypergeometric series. One of them yields in a direct limit Bailey's celebrated ${}_6ψ_6$ summation formula, answering a question recently raised by Chen and Fu ({\em Semi-Finite Forms of Bilateral Basic Hypergeometric Series}, Proc. Amer. Math. Soc., to appear).
We present a new matrix inverse with applications in the theory of bilateral basic hypergeometric series. Our matrix inversion result is directly extracted from an instance of Bailey's very-well-poised 6-psi-6 summation theorem, and involves two infinite matrices which are not lower-triangular. We combine our bilateral matrix inverse with known basic hypergeometric summation theorems to derive, via inverse relations, several new identities for bilateral basic hypergeometric series.
We show that several classical bilateral summation and transformation formulas have semi-finite forms. We obtain these semi-finite forms from unilateral summation and transformation formulas. Our method can be applied to derive Ramanujan's $_1ψ_1$ summation, Bailey's $_2ψ_2$ transformations, and Bailey's $_6ψ_6$ summation.
Theta functions and transformations of bilateral basic hypergeometric series
本文通过theta函数和双侧基本超几何级数建立新变换公式，构造伴生q级数，证明McIntosh关于q级数渐近变换的猜想，并扩展了Ramanujan和McIntosh的一些恒等式。
In terms of the analytic continuation method, we prove three transformation formulas involving bilateral basic hypergeometric series. One of them is equivalent to Jouhet's result involving two $_8ψ_8$ series and two $_8ϕ_7$ series.
In this paper, we prove several transformation formulas for the very-well-poised bilateral basic hypergeometric $_5ψ_5$ series by using the relationship between the bilateral basic hypergeometric $_5ψ_5$ series and basic hypergeometric $_8ϕ_7$ series.
In this paper, we establish two new transformation formulas for ${}_{8}ψ_{8}$ and ${}_8ϕ_7$ series by means of Slater's general transformation for bilateral series. As applications, some specific transformation formulas are presented among which include a general form of Weierstrass' theta identity and new proofs of Bailey's VWP ${}_6ψ_6$ and Jackson's ${}_8ϕ_7$ summation formula.
A general $q$-series transformation and its applications to multi-sum Rogers-Ramanujan-Slater identities
本文提出一个包含 Nahm 级数变换的一般 $q$-级数变换，并由此导出 ${}_{r+1}ϕ_r$ 级数的新变换公式及多和 Rogers-Ramanujan-Slater 型 $q$-恒等式。
A $q$-Exponential Operator Based on the Derivative of Order 1 and Summation of Bilateral Basic Hypergeometric Series
本文提出一种基于一阶q^{±1}导数的q-指数算子，用于推导多种双边基本超几何级数的求和公式。
Bilateral Two-Parameter Mock Theta Functions and Related Applications
本文研究双边双参数模拟 theta 函数，建立双边级数新关系，导出双边模拟 theta 函数恒等式，并总结经典模拟 theta 函数与双边级数的关系。
Asymmetric bilateral Bailey pairs and Rogers-Ramanujan type identities
本文通过非对称双侧Bailey对推导出罗杰斯-拉马努金型、安德鲁斯-戈登型及假θ函数恒等式，并基于Dousse等人构建的Bailey晶格得到非对称双侧Bailey引理，进一步推导出Appell-Lerch级数恒等式。
Separable integer partition classes and Slater's list -- I
本文利用Andrews的可分离整数划分理论，对Slater列表中的若干恒等式进行解析，提供自然的划分解释和参数化推广，并通过q-超几何变换推导出新的表达式。
Separable integer partition classes and Slater's list -- II
应用Andrews的可分离整数划分理论，构造了严格超划分和带位置间隙条件的超划分族，生成函数精细化了Slater恒等式的级数侧，并通过变换得到新恒等式。
A family of orthogonal functions on the unit circle and a new multilateral matrix inverse
本文利用Bailey求和证明了特定级数构成单位圆上的正交函数族，并从Dougall求和提取出新的双边矩阵逆，结合Pfaff-Saalschütz求和推导了 $_3H_3$ 级数求和公式及其多变量扩展。
We set up a combinatorial framework for inclusion-exclusion on the partitions into distinct parts to obtain an alternative generating function of partitions into distinct and non-consecutive parts. In connection with Rogers-Ramanujan identities, the generating function yields two formulas in Slater's list. The same formulas were constructed by Hirschhorn. Similar formulas were obtained by Bringmann, Mahlburg and Nataraj. We also use staircases to give alternative triple series for partitions into $d-$distinct parts for any $d \geq 2$.
Bilateral truncated quintuple product
该研究建立五重乘积恒等式的双边截断形式，推广已有截断恒等式，并给出两个相关 q-级数恒等式的双边截断版本。
We generalize a terminating summation formula to a unilateral nonterminating, and further, a bilateral summation formula by a property of analytic functions. The unilateral one is proved to be a $q$-analogue of a $_4F_3$-summation formula. And, an identity unifying Jacobi's triple product identity and the quintuple product identity is obtained as a special case of the bilateral one.
In [Electron. J. Combin. 10 (2003), #R10], the author presented a new basic hypergeometric matrix inverse with applications to bilateral basic hypergeometric series. This matrix inversion result was directly extracted from an instance of Bailey's very-well-poised 6-psi-6 summation theorem, and involves two infinite matrices which are not lower-triangular. The present paper features three different multivariable generalizations of the above result. These are extracted from Gustafson's A_r and C_r extensions and of the author's recent A_r extension of Bailey's 6-psi-6 summation formula. By combi…
Using a simple classical method we derive bilateral series identities from terminating ones. In particular, we show how to deduce Ramanujan's 1-psi-1 summation from the q-Pfaff-Saalschuetz summation. Further, we apply the same method to our previous q-Abel-Rothe summation to obtain, for the first time, Abel-Rothe type generalizations of Jacobi's triple product identity. We also give some results for multiple series.
In this we paper we prove several new identities of the Rogers-Ramanujan-Slater type. These identities were found as the result of computer searches. The proofs involve a variety of techniques, including series-series identities, Bailey pairs, a theorem of Watson on basic hypergeometric series, generating functions and miscellaneous methods.
We prove a new Fermionic quasiparticle sum expression for the character of the Ising model vertex algebra, related to the Jackson-Slater $q$-series identity of Rogers-Ramanujan type and to Nahm sums for the matrix $\left( \begin{smallmatrix} 8 & 3 \\ 3 & 2 \end{smallmatrix} \right)$. We find, as consequences, an explicit monomial basis for the Ising model, and a description of its singular support. We find that the ideal sheaf of the latter, defining it as a subscheme of the arc space of its associated scheme, is finitely generated as a differential ideal. We prove three new $q$-series identit…
A complete characterization of two functions $f(x,y)$ and $g(x,y)$ in the $(f,g)$-inversion is presented. As an application to the theory of hypergeometric series, a general bibasic summation formula determined by $f(x,y)$ and $g(x,y)$ as well as four arbitrary sequences is obtained which unifies Gasper and Rahman's, Chu's and Macdonald's bibasic summation formula. Furthermore, an alternative proof of the $(f,g)$-inversion derived from the $(f,g)$-summation formula is presented. A bilateral $(f,g)$-inversion containing Schlosser's bilateral matrix inversion as a special case is also obtained.
We give new proofs for certain bilateral basic hypergeometric summation formulas using the symmetries of the corresponding series. In particular, we present a proof for Bailey's $_3ψ_3$ summation formula as an application. We also prove a multiple series analogue of this identity by considering hyperoctahedral group symmetries of higher ranks.
We give elementary derivations of some classical summation formulae for bilateral (basic) hypergeometric series. In particular, we apply Gauss' 2-F-1 summation and elementary series manipulations to give a simple proof of Dougall's 2-H-2 summation. Similarly, we apply Rogers' nonterminating 6-phi-5 summation and elementary series manipulations to give a simple proof of Bailey's very-well-poised 6-psi-6 summation. Our method of proof extends M. Jackson's first elementary proof of Ramanujan's 1-psi-1 summation.
In this article, we derive some identities for multilateral basic hypergeometric series associated to the root system A_n. First, we apply Ismail's argument to an A_n q-binomial theorem of Milne and derive a new A_n generalization of Ramanujan's 1-psi-1 summation theorem. From this new A_n 1-psi-1 summation and from an A_n 1-psi-1 summation of Gustafson we deduce two lemmas for deriving simple A_n generalizations of bilateral basic hypergeometric series identities. These lemmas are closely related to the Macdonald identities for A_n. As samples for possible applications of these lemmas, we pro…
Some $q$-transformation formulas and Rogers-Ramanujan type identities
本文利用 Liu 变换公式结合其他 $q$-级数求和，推导参数化恒等式，并从中获得 75 个已知及若干新的 Rogers-Ramanujan 型恒等式。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。