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 Ae or symplectic Q groups. The classical case, corresponding to A¡ or equivalently U(2), contains an immense amount of the theory and application of one-variable basic hypergeometric series, includin...

In this paper, we derive explicit product formulas and positive convolution structures for three continuous classes of Heckman-Opdam hypergeometric functions of type BC. For specific discrete series of multiplicities these hypergeometric functions occur as the spherical functions of non-compact Grassmann manifolds G/K over one of the (skew) fields F = R,C,H. We write the product formula of thes...

The “Diophantine” properties of the zeros of certain polynomials in the Askey scheme, recently discovered by Calogero and his collaborators, are explained, with suitably chosen parameter values, in terms of the summation theorem of hypergeometric series. Here the Diophantine property refers to integer valued zeros. It turns out that the same procedure can also be applied to polynomials arising ...

We give new proofs and explain the origin of several combinatorial identities of Fu and Lascoux, Dilcher, Prodinger, Uchimura, and Chen and Liu. We use the theory of basic hypergeometric functions, and generalize these identities. We also exploit the theory of polynomial expansions in the Wilson and Askey-Wilson bases to derive new identities which are not in the hierarchy of basic hypergeometr...

We consider the q-hypergeometric equation with q = 1 and α, β, γ ∈ Z. We solve this equation on the space of functions given by a power series multiplied by a power of the logarithmic function. We prove that the subspace of solutions is twodimensional over the field of quasi-constants. We get a basis for this space explicitly. In terms of this basis, we represent the q-hypergeometric function o...

In a series of papers, we discussed the solution of Laplace’s differential equation (DE) by using fractional calculus, operational calculus in the framework of distribution theory, and Laplace transform. The solutions of Kummer’s DE, which are expressed by the confluent hypergeometric functions, are obtained with the aid of the analytic continuation (AC) of Riemann–Liouville fractional derivati...

Some examples of naturally arising multisum q-series which turn out to have representations as fermionic single sums are presented. The resulting identities are proved using transformation formulas from the theory of basic hypergeometric series.

Using hypergeometric identities and certain representations for Eisenstein series, we uniformly derive several new series representations for 1/ 2. © 2008 Elsevier Inc. All rights reserved. MSC: 33C05; 33E05; 11F11; 11R29

The left hand side (or A–side) of this identity was a generating series for the numbers n(d) of rational curves of various degrees d lying on a smooth quintic hypersurface in P. The right hand side (B–side) was a certain hypergeometric function. The Mirror Identity states that the two functions become identical after an explicit change of variables which is defined as a quotient of two hypergeo...

Abstract. A proof of an unusual summation formula for a basic hypergeometric series associated to the affine root system Ãn that was conjectured byWarnaar is given. It makes use of Milne’s An extension of Watson’s transformation, Ramanujan’s 1ψ1summation, and a determinant evaluation of the author. In addition, a transformation formula between basic hypergeometric series associated to the affin...