Nonpolynomial basic hypergeometric eigenfunctions of the Askey–Wilson second order difference operator are known to be expressible as very-well-poised 8φ7 series. In this paper we use this fact to derive various basic hypergeometric and theta function identities. We relate most of them to identities from the existing literature on basic hypergeometric series. This leads for example to a new der...

It is well known that many bilateral basic hypergeometric identities can be derived from unilateral identities. Using Cauchy’s method [5, 15, 20, 21] one may obtain bilateral basic hypergeometric identities from terminating unilateral identities. Starting with nonterminating unilateral basic hypergeometric series, Chen and Fu [8] developed a method to construct semi-finite forms by shifting the...

Structures of symmetries of transformations for Holman–Biedenharn–Louck An hypergeometric series: An terminating balanced 4F3 series and An elliptic 10E9 series are discussed. Namely the description of the invariance groups and the classification all of possible transformations for each types of An hypergeometric series are given. Among them, a “periodic” affine Coxeter group which seems to be ...

This survey article (which will appear as a chapter in the book “Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions”, Springer-Verlag) provides a small collection of basic material on multiple hypergeometric series of Appell-type and of more general series of related type.

The Abel method on summation by parts is reformulated to present new and elementary proofs of several classical identities of terminating balanced basic hypergeometric series. The examples strengthen our conviction that as traditional analytical instrument, the revised Abel method on summation by parts is indeed a very natural choice for working with basic hypergeometric series.

when the relation B−A+C = 1 holds. In this paper a formula is presented which evaluates this series in case when B −A+C is an integer. The formula expresses the infinite series as a linear combination of two Γ-terms with coefficients being finite hypergeometric 3F2 series. Algorithmical problems of summation of infinite hypergeometric series are considered in the light of the generalized formul...

which holds for mi non-negative integers and Re (a + |m|) < 1. Accordingly, hypergeometric series with integral parameter differences have been called Karlsson–Minton type hypergeometric series; other results for such series may be found in [C, G1, G2, S1, S2]. In recent work [R], we have derived a very general reduction formula for series of Karlsson–Minton type. We recall it in its most gener...

We give a method to embed the q-series in a (p, q)-series and derive the corresponding (p, q)-extensions of the known q-identities. The (p, q)-hypergeometric series, or twin-basic hypergeometric series (different from the usual bibasic hypergeometric series), is based on the concept of twin-basic number [n]p,q = (p n−qn)/(p−q). This twin-basic number occurs in the theory of two-parameter quantu...