Identities between q-hypergeometric and hypergeometric integrals of different dimensions

Bilinear Identity for q - Hypergeometric Integrals

Theta Hypergeometric Integrals

A general class of (multiple) hypergeometric type integrals associated with the Jacobi theta functions is defined. These integrals are related to theta hypergeometric series via the residue calculus. In the one variable case, theta function extensions of the Meijer function are obtained. A number of multiple generalizations of the elliptic beta integral associated with the root systems An and C...

Limits of elliptic hypergeometric integrals

In [16], the author proved a number of multivariate elliptic hypergeometric integrals. The purpose of the present note is to explore more carefully the various limiting cases (hyperbolic, trigonometric, rational, and classical) that exist. In particular, we show (using some new estimates of generalized gamma functions) that the hyperbolic integrals (previously treated as purely formal limits) a...

A new elementary algorithm for proving q-hypergeometric identities

We give a fast elementary algorithm to get a small number n1 for an admissible q-properhypergeometric identity ∑ k F(n, k) = G(n), n ≥ n0 such that we can prove the identity by checking its correctness for n (n0 ≤ n ≤ n1). For example, we get n1 = 191 for the q-Vandermonde-Chu identity, n1 = 70 for a finite version of Jacobi’s triple product identity and n1 = 209 for an identity due to L.J. Rog...

qMultiSum--a package for proving q-hypergeometric multiple summation identities

A Mathematica package for finding recurrences for q-hypergeometric multiple sums is introduced. Together with a detailed description of the theoretical background, we present several examples to illustrate its usage and range of applicability. In particular, various computer proofs of recently discovered identities are exhibited.