We deduce new q-series identities by applying inverse relations to certain identities for basic hypergeometric series. The identities obtained themselves do not belong to the hierarchy of basic hypergeometric series. We extend two of our identities, by analytic continuation, to bilateral summation formulae which contain Ramanujan’s 1ψ1 summation and a very-well-poised 4ψ6 summation as special c...

We complete the construction of raising and lowering operators, given in a previous work, for the orthogonal polynomials of hypergeometric type on nonhomogeneous lattice, and extend these operators to the generalized orthogonal polynomials, namely, those difference of orthogonal polynomials that satisfy a similar difference equation of hypergeometric type. PACS Numbers: 0210N, 0220S, 0230V, 027...

We undertake the study of bivariate Horn systems for generic parameters. We prove that these hypergeometric systems are regular holonomic, and we provide an explicit formula for their holonomic rank as well as bases of their spaces of complex holomorphic solutions. We also obtain analogous results for the generalized hypergeometric systems arising from lattices of any rank.

Recently, Bringmann and Kane established two new Bailey pairs and used them to relate certain q-hypergeometric series to real quadratic fields. We show how these pairs give rise to new mock theta functions in the form of q-hypergeometric double sums. Additionally, we prove an identity between one of these sums and two classical mock theta functions introduced by Gordon and McIntosh.

In his 1984 Ph.D. thesis, J. Greene defined an analogue of the Euler integral transform for finite field hypergeometric series. Here we consider a special family of matrices which arise naturally in the study of this transform and prove a conjecture of Ono about the decomposition of certain finite field hypergeometric functions into functions of lower dimension.

The main goal of our survey is to give common characteristics of auxiliary hypergeometric functions (and their generalisations), functions which occur in number-theoretical problems. Originally designed as a tool for solving these problems, the hypergeometric series have become a connecting link between different areas of number theory and mathematics in general. Bibliography: 183 titles.

We consider bases of Pfaffian systems for A-hypergeometric systems. These are given by Gröbner deformations, they also provide bases for twisted cohomology groups. For a hypergeometric system associated with a class of order polytopes, these bases have a combinatorial description. The size of the bases associated with a subclass of the order polytopes has a growth rate of polynomial order.

Abstract. 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 obtain further Bailey-type transformations as consequences of our result, some of which are new also in the case of basic and classical hypergeometric s...

1. Frey representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 1.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 1.2. Classification: The rigidity method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 1.3. Construction: Hyper...

We show that many integrals containing products of confluent hypergeometric functions follow directly from one single integral that has a very simple formula in terms of Appell’s double series F2 . We present some techniques for computing such series. Applications requiring the matrix elements of singular potentials and the perturbed Kratzer potential are presented. PACS 03.65.Ge Integrals cont...