1 I. Waller, Zeit. Physik, 38, 644 (1926). 2 J. H. Van Vleck, Proc. Roy. Soc. A, 143, 679 (1934). There is a slight error in his expression for r for 1=2, the coefficients being all 10 times too large. 3a may be equated to zero except in the case q = 21. 4 W. N. Bailey, Generalized Hypergeometric Series (Cambridge Tract No. 32), p. 18. 6 The polynomial 1n(Z) is discussed by H. Bateman, T8kohu M...

We prove a summation formula for a bilateral series whose terms are products of two basic hypergeometric functions. In special cases, series of this type arise as matrix elements of quantum group representations.

I will discuss some identities for generalized hypergeometric series that were discovered quite recently in connection with rational approximations to π, π2 and π4. A curious thing is that most of these identities have “automatic” proofs (using creative telescoping), and a problem is to provide “human” proofs by means of classical hypergeometric summation and transformation formulas. Let me say...

Abstract We derive a duality formula for two-row Macdonald functions by studying their relation with basic hypergeometric functions. We introduce two parameter vertex operators to construct a family of symmetric functions generalizing Hall-Littlewood functions. Their relation with Macdonald functions is governed by a very well-poised q-hypergeometric functions of type 4 0$, for which we obtain ...

We prove a multivariable elliptic extension of Jackson’s summation formula conjectured by Spiridonov. The trigonometric limit case of this result is due to Gustafson and Rakha. As applications, we obtain two further multivariable elliptic Jackson summations and two multivariable elliptic Bailey transformations. The latter four results are all new even in the trigonometric case.

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

We give a Newton type rational interpolation formula (Theorem 2.2). It contains as a special case the original Newton interpolation, as well as the recent interpolation formula of Zhi-Guo Liu, which allows to recover many important classical q-series identities. We show in particular that some bibasic identities are a consequence of our formula.

Abstract. In examining the relationship between the number of points over Fp on certain Calabi-Yau manifolds and hypergeometric series which correspond to a particular period of the manifold, Rodriguez-Villegas identified 22 possible supercongruences. We provide a framework of congruences covering all 22 cases. Using this framework we prove one of the outstanding supercongruence conjectures bet...

Mathematicians often speak of the unity of their subject. Such a unity from the 19th century origins in the history of linear ordinary differential equations, especially those closely connected with elliptic and modular functions. It is concerned with the impact of group theoretical and geometrical ideas upon the problem of understanding the nature of the solutions to a differential equation (s...

Abstract. We announce a higher-dimensional generalization of the Bailey Transform, Bailey Lemma, and iterative “Bailey chain” concept in the setting of basic hypergeometric series very well-poised on unitary Al or symplectic Cl groups. The classical case, corresponding to A1 or equivalently U(2), contains an immense amount of the theory and application of one-variable basic hypergeometric serie...