Related papers: A simple proof of Bailey's very-well-poised 6-psi-…
This paper gives a short but reasonably comprehensive review of Ramanujan's {_1\psi_1} summation and its generalisations. It covers the history of Ramanujan's summation, simple applications to sums of squares and orthogonal polynomials,…
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.…
We extend expansion formulas of Liu given in 2013 to the context of multiple series over root systems. Liu and others have shown the usefulness of these formulas in Special Functions and number-theoretic contexts. We extend Wang and Ma's…
By systematically applying ten inequivalent two-part relations between hypergeometric sums 3F2(1) to the published database of all such sums, 66 new sums are obtained. Many results extracted from the literature are shown to be special cases…
In terms of the analytic continuation method, we give the united proofs for three $q$-extensions of Dougall's $_2H_2$ summation formula. Some related results are also discussed in this paper.
By systematically applying ten well-known and inequivalent two-part relations between hypergeometric sums 3F2(...|1) to the published database of all such sums, 62 new sums are obtained. The existing literature is summarized, and many…
Gauss hypergeometric functions with a dihedral monodromy group can be expressed as elementary functions, since their hypergeometric equations can be transformed to Fuchsian equations with cyclic monodromy groups by a quadratic change of the…
We describe the utility of integral representations for sums of basic hypergeometric functions. In particular we use these to derive an infinite sequence of transformations for symmetrizations over certain variables which the functions…
The aim of this paper is to establish new series transforms of Bailey type and to show that these Bailey type transforms work as efficiently as the classical one and give not only new $q$-hypergeometric identities, converting double or…
The partial sums of two quartic basic hypergeometric series are investigated by means of the modified Abel lemma on summation by parts. Several summation and transformation formulae are consequently established.
We give a definition of generalized hypergeometric functions over finite fields using modified Gauss sums, which enables us to find clear analogy with classical hypergeometric functions over the complex numbers. We study their fundamental…
Six families of generalized hypergeometric series in a variable $x$ and an arbitrary number of parameters are considered. Each of them is indexed by an integer $n$. Linear recurrence relations in $n$ relate these functions and their product…
Recursive formulas extending some known $_{2}F_{1}$ and $_{3}F_{2}$ summation formulas by using contiguous relations have been obtained. On the one hand, these recursive equations are quite suitable for symbolic and numerical evaluation by…
An elementary proof is given for a nonterminating "strange" cubic $_7F_6$-series summation formula of Gasper and Rahman, through the modified Abel lemma on summation by parts. As a byproduct, an interesting nonterminating…
By employing certain extended classical summation theorems, several surprising \pi and other formulae are displayed.
We give an overview of some of the main results from the theories of hypergeometric and basic hypergeometric series and integrals associated with root systems. In particular, we list a number of summations, transformations and explicit…
We present sum-sides for principal characters of all standard (i.e., integrable and highest-weight) irreducible modules for the affine Lie algebra $A_2^{(2)}$. We use modifications of five known Bailey pairs; three of these are sufficient…
Using some properties of the gamma function and the well-known Gauss summation formula for the classical hypergeometric series, we prove a four-parameter series expansion formula, which can produce infinitely many Ramanujan type series for…
Recently, Kajihara gave a Bailey-type transformation relating basic hypergeometric series on the root system An, with different dimensions n. We give, with a new, elementary, proof, an elliptic analogue of this transformation. We also…
A summation formula is derived for the hypergeometric series of unit argument ${}_3F_2(1,1,c;d,n+2;1)$, where $n=0, 1, 2, \ldots$ and $\Re (d-c+n)>0$.