paper

Identities between q-hypergeometric and hypergeometric integrals of different dimensions

arXiv:math/0309372

Abstract

Given complex numbers $m_{1}, l_{1}$ and nonnegative integers $m_{2}, l_{2}$ , such that $m_{1} + m_{2} = l_{1} + l_{2}$ , for any $a, b = 0, ..., min (m_{2}, l_{2})$ we define an $l_{2}$ -dimensional Barnes type q-hypergeometric integral $I_{a, b} (z, I^¼; m_{1}, m_{2}, l_{1}, l_{2})$ and an $l_{2}$ -dimensional hypergeometric integral $J_{a, b} (z, I^¼; m_{1}, m_{2}, l_{1}, l_{2})$ . The integrals depend on complex parameters $z$ and $I^¼$ . We show that $I_{a, b} (z, I^¼; m_{1}, m_{2}, l_{1}, l_{2})$ equals $J_{a, b} (e^{I^} ¼, z; l_{1}, l_{2}, m_{1}, m_{2})$ up to an explicit factor, thus establishing an equality of $l_{2}$ -dimensional q-hypergeometric and $m_{2}$ -dimensional hypergeometric integrals. The identity is based on the $(g l_{k}, g l_{n})$ duality for the qKZ and dynamical difference equations.

Preprint (2003), 14 pages, AmsLaTeX, references updated