We use elliptic Taylor series expansions and interpolation to deduce a number of summations for elliptic hypergeometric series. We extend to the well-poised elliptic case results that in the $q$-case have previously been obtained by Cooper and by Ismail and Stanton. We also provide identities involving S. Bhargava's cubic theta functions.
In this paper we derive the infinite summation formulas of Srivastava's general triple hypergeometric function. Certain particular cases leading to infinite summation formulas for fourteen Lauricella and three Srivastavaś triple hypergeometric functions are also presented.
Multilateral inversion of A_r, C_r and D_r basic hypergeometric series
在[Electron. J. Combin. 10（2003），#R10]的基础上，提出了一种新的基本超几何矩阵逆，并应用于双边基本超几何级数.这个矩阵求逆结果直接从Bailey的非常好的平衡6-psi-6求和定理的一个例子中提取，并且涉及两个不是下三角形的无限矩阵。本文的特点是上述结果的三个不同的多变量推广。它们是从Gustafson的A_r和C_r推广以及作者最近对Bailey的6-psi-6求和公式的A_r推广中抽取出来的。将这些新的多维矩阵逆与Jackson的8-phi-7求和定理的A_r和D_r推广相结合，得到了与根系A_r和C_r有关的三个平衡的非常平衡的8-psi-8求和定理。
Multidimensional matrix inversions provide a powerful tool for studying multiple hypergeometric series. In order to extend this technique to elliptic hypergeometric series, we present three new multidimensional matrix inversions. As applications, we obtain a new $A_r$ elliptic Jackson summation, as well as several quadratic, cubic and quartic summation formulas.
The main aim of this paper is to derive some new summation theorems for terminating and truncated Clausen's hypergeometric series with unit argument, when one numerator parameter and one denominator parameter are negative integers. Further, using our truncated summation theorems, we obtain the Mellin transforms of the product of exponential function and Goursat's truncated hypergeometric function.
In 2010, Kh. Hessami Pilehrood and T. Hessami Pilehrood introduced generating function identities used to obtain series accelerations for values of Dirichlet's $β$ function, via the Markov--Wilf--Zeilberger method. Inspired by these past results, together with related results introduced by Chu et al., we introduce a variety of hypergeometric recurrences. We prove these recurrences using the WZ method, and we apply these recurrences to obtain series acceleration identities. We introduce a family of summations generalizing a Ramanujan-type series for $\frac{1}{π^2}$ due to Guillera, and a family…
Some more semi-finite forms of bilateral basic hypergeometric series
证明了双边基本超几何级数的一些新的半有限形式。其中之一在直接极限中产生Bailey著名的${}_6ψ_6$求和公式，回答了Chen和Fu最近提出的一个问题（{ \em Semi-Finite Forms of Bilateral Basic超几何级数}，Proc. Amer. Math. Soc.，出现）。
In 1981, Andrews gave a four-variable generalization of Ramanujan's ${_1ψ_1}$ summation formula. We establish a six-variable generalization of Andrews' identity according to the transformation formula for two ${_8ϕ_7}$ series and Bailey's transformation formula for three ${_8ϕ_7}$ series. Then it is used to find a six-variable generalization of Ramanujan's reciprocity theorem, which is different from Liu's formula. We derive the generalizations of Bailey's two $_3ψ_3$ summation formulas in terms of two limiting relations and Bailey's another transformation formula for three $_8ϕ_7$ series. Bas…
In this brief note, we show how to apply Kummer's and other quadratic transformation formulas for Gauss' and generalized hypergeometric functions in order to obtain transformation and summation formulas for series with harmonic numbers that contain one or two continuous parameters. We also give a generating function of the sequence $\frac{(a)_n (1-a)_n}{(n!)^2}H_n$ as a combination of Gauss hypergeometric function and elementary functions.
In our recent work we proposed a generalization of the beta integral method for derivation of the hypergeometric identities which can by analogy be termed "the G function integral method". In this paper we apply this technique to the cubic and the degenerate Miller-Paris transformations to get several new transformation and summation formulas for the generalized hypergeometric functions at a fixed argument. We further present an alternative approach for reducing the right hand sides resulting from our method to a single hypergeometric function which does not require the use of summation formul…
