Certain summation and transformation formulas for generalized hypergeometric series

We derive summation formulas for generalized hypergeometric series of unit argument, one of which upon specialization reduces to Minton’s summation theorem. As an application we deduce a reduction formula for a certain Kampé de Fériet function that in turn provides a Kummer-type transformation formula for the generalized hypergeometric function _pF_p(x).     

More Semi-Finite Forms of Bilateral Basic Hypergeometric Series

We prove some new semi-finite forms of bilateral basic hypergeometric series. One of them yields in a direct limit Bailey’s celebrated _6ψ6 summation formula, answering a question recently raised by Chen and Fu. Received November 17, 2005     

An alternative proof of the extended Saalschütz summation theorem for the r + 3Fr + 2(1) series with applications

A simple proof is given of a new summation formula recently added in the literature for a terminating _{r + 3}F_{r + 2}(1) hypergeometric series for the case when r pairs of numeratorial and denominatorial parameters differ by positive integers. This formula represents an extension of the well‐known Saalschütz summation formula for a ₃F₂(1) series. Two applications of this extended summation formula are discussed. The first application extends two identities given by Ramanujan and the second, which also employs a similar extension of the Vandermonde–Chu summation theorem for the ₂F₁ series, extends certain reduction formulas for the Kampé de Fériet function of two variables given by Exton and Cvijovi? & Miller. Copyright © 2015 John Wiley & Sons, Ltd.     

Dn Basic Hypergeometric Series

We study multiple series extensions of basic hypergeometric series related to the root system D_n. We make a small change in the notation used for C_n and D_n series to bring them closer to A_n series. This allows us to combine the three types of series, and get D_n extensions of the following classical summation and transformation theorems: The q-Pfaff-Saalschütz summation, Rogers' _{6 5} sum, the q-Gauss summation, q-Chu-Vandermonde summations, Watson's q-analogue of Whipple's transformation, and the q-Dougall summation theorem. We also define A_n and C_n extensions of the Rogers-Selberg function, and prove a reduction formula for both of them. This generalizes some work of Andrews. We use some techniques originally developed to study multiple basic hypergeometric series related to the root system A_n (U(n + 1) basic hypergeometric series).     

Elliptic hypergeometric series on root systems

We derive a number of summation and transformation formulas for elliptic hypergeometric series on the root systems A_n, C_n and D_n. In the special cases of classical and q-series, our approach leads to new elementary proofs of the corresponding identities.     

Transformation formula for a double Clausenian hypergeometric series,its q-analogue,and its invariance group

A transformation formula for a double basic hypergeometric series of type Φ_0:2;2^1:2;2 is derived. This transformation yields a double series analogue of Sears’ transformation for a terminating ₃Φ₂ series. In the limit q→1, the formula reduces to a transformation for a terminating double Clausenian hypergeometric series of unit argument (one of the proper Kampé de Fériet series, F_0:2;2^1:2;2(1,1)). This formula is a double series analogue of Whipple's terminating ₃F₂ transformation. This transformation gives rise to a transformation group (the invariance group) acting on the parameters of the double series. The invariance group is examined and shown to be a subgroup of a double copy of the symmetries of the square.     

Abel's method on summation by parts and terminating well-poised q-series identities

The Abel method on summation by parts is reformulated to present new and elementary proofs of several classical identities of terminating well-poised basic hypergeometric series, mainly discovered by [F H. Jackson, Certain q-identities, Quart. J. Math. Oxford Ser. 12 (1941) 167–172]. This strengthens further our conviction that as a traditional analytical instrument, the revised Abel method on summation by parts is indeed a very natural choice for working with basic hypergeometric series.     

A Multidimensional Generalization of Shukla's 8ψ8 Summation

Abstract. We give an r -dimensional generalization of H. S. Shukla's very-well-poised ₈ ψ ₈ summation formula. We work in the setting of multiple basic hypergeometric series very-well-poised over the root system A _r-1 or, equivalently, the unitary group U(r) . Our proof, which is already new in the one-dimensional case, utilizes an A _r-1 nonterminating very-well-poised ₆ φ ₅ summation by S. C. Milne, a partial fraction decomposition, and analytic continuation.     

Transformation and summation formulas for double hypergeometric series

A number of new transformation formulas for double hypergeometric series are presented. The series appearing here are the so-called Kampé de Fériet functions of type F_1:1;2^0:3;4(1,1) and F_0:2;2^1:2;2(1,1). The transformation formulas relate such double series to a single hypergeometric series of ₄F₃(1) type. By specializing certain parameters, a list of new summation formulas for F_0:2;2^1:2;2(1,1) series is obtained. The origin of the results comes from studying symmetries of the 9-j coefficient appearing in quantum theory of angular momentum.     

Applications of operator identities to the multiple q-binomial theorem and q-Gauss summation theorem

In this paper, we first give an interesting operator identity. Furthermore, using the q-exponential operator technique to the multiple q-binomial theorem and q-Gauss summation theorem, we obtain some transformation formulae and summation theorems of multiple basic hypergeometric series.     

The Saalschütz chain reactions and bilateral basic hypergeometric series

By iterating recursively the q-Saalschütz summation formula, we introduce the Saalschütz chain reactions. A general series transform, which expresses a nonterminating bilateral series in terms of a finite multiple unilateral sum, will be established. As applications we derive, by means of Bailey’s 6ψ6 -series identity, several bilateral transformations including one due to Milne [12]. These transformations further yield a number of closed formulas of very well-poised bilateral basic hypergeometric series; which are closely related to the identities obtained by Minton [13], Karlsson [11], Gasper [8], and Chu [5], [6], [7] through the partial fraction method and divided differences.     

A q-analog of the 5F4(1) summation theorem for hypergeometric series well-poised in SU(n)

As an application of a general q-difference equation for basic hypergeometric series well-poised in SU(n), an elementary proof is given of a q-analog of Holman's SU(n) generalization of the terminating ₅F₄(1) summation theorem. This provides an SU(n) generalization of the terminating ₆Φ₅ summation theorem for classical basic hypergeometric series.     

Some families of combinatorial and other series identities and their applications

The main object of this presentation is to show how some simple combinatorial identities can lead to several general families of combinatorial and other series identities as well as summation formulas associated with the Fox-Wright function _pΨ_q and various related generalized hypergeometric functions. At least one of the hypergeometric summation formulas, which is derived here in this manner, has already found a remarkable application in producing several interesting generalizations of the Karlsson-Minton summation formula. We also consider a number of other combinatorial series identities and rational sums which were proven, in recent works, by using different methods and techniques. We show that much more general results can be derived by means of certain summation theorems for hypergeometric series. Relevant connections of the results presented here with those in the aforementioned investigations are also considered.     

The Cn WP-Bailey chain

The purpose of this paper is to introduce the concept of C_n WP-Bailey pairs. The C_n WP-Bailey transform is obtained by applying the Cn 6φ5 summation formula. From this result, the Cn WP-Bailey lemma is deduced by making use of the Cn q-Dougall summation formula. Some applications are investigated. Finally, the case of elliptic Cn WP-Bailey pairs is discussed.     

A new two-term relation for the 3F2 hypergeometric function of unit argument

The elementary manipulation of series together with summations of Gauss, Saalschutz and Dixon are employed to deduce a two-term relation for the hypergeometric function ₃F₂(1) and a summation formula for the same function, neither of which has previously appeared in the literature. The two-term relation has implications in the calculus of finite differences.     

A Multidimensional Generalization of Shukla's 8ψ8 Summation

   Abstract. We give an r -dimensional generalization of H. S. Shukla's very-well-poised ₈ ψ ₈ summation formula. We work in the setting of multiple basic hypergeometric series very-well-poised over the root system A _r-1 or, equivalently, the unitary group U(r) . Our proof, which is already new in the one-dimensional case, utilizes an A _r-1 nonterminating very-well-poised ₆ φ ₅ summation by S. C. Milne, a partial fraction decomposition, and analytic continuation.     

Two new transformation formulas of basic hypergeometric series

By means of a modified version of Cauchy's method for obtaining bilateral series identities, two new transformation formulas for bilateral basic hypergeometric series are derived. These contain several important identities for basic hypergeometric series as special cases, including the nonterminating q-Saalschütz summation, Bailey's very well-poised summation and the nonterminating Watson transformation.     

Elementary derivations of identities for bilateral basic hypergeometric series

We give elementary derivations of several classical and some new summation and transformation formulae for bilateral basic hypergeometric series. For motivation, we review our previous simple proof (Proc. Amer. Math. Soc. 130 (2002), 1103-1111) of Bailey's very-well-poised ₆y₆_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 ₈y₈_8\psi_8 transformation, a very-well-poised ₁₀y₁₀_{10}\psi_{10} transformation, by induction, Slater's general transformation for very-well-poised _2ry_2r_{2r}\psi_{2r} series, and Slater's transformation for general _ry_r_{r}\psi_{r} series. Finally, we derive some new transformations for bilateral basic hypergeometric series of a specific type.     

Bailey type summation formulas associated with the root system $G_{2}^{\vee}$

We present three types of summation formulas for the root system G₂^úG_{2}^{\vee}, which are generalized from Bailey’s summation formula for a very-well-poised balanced ₆ ψ ₆ basic hypergeometric series.     

Method of summation of some slowly convergent series

A new method of summation of slowly convergent series is proposed. It may be successfully applied to the summation of generalized and basic hypergeometric series, as well as some classical orthogonal polynomial series expansions. In some special cases, our algorithm is equivalent to Wynn’s epsilon algorithm, Weniger transformation [E.J. Weniger, Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series, Computer Physics Reports 10 (1989) 189-371] or the technique recently introduced by ?í?ek et al. [J. ?í?ek, J. Zamastil, L. Skála, New summation technique for rapidly divergent perturbation series. Hydrogen atom in magnetic field, Journal of Mathematical Physics 44 (3) (2003) 962-968]. In the case of trigonometric series, our method is very similar to the Homeier’s H transformation, while in the case of orthogonal series — to the K transformation. Two iterated methods related to the proposed method are considered. Some theoretical results and several illustrative numerical examples are given.