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29 March 2024
 
  » arxiv » math.QA/0309372

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Identities between q-hypergeometric and hypergeometric integrals of different dimensions
V.Tarasov ; A.Varchenko ;
Date 23 Sep 2003
Subject Quantum Algebra; Mathematical Physics; Representation Theory | math.QA math-ph math.MP math.RT
AbstractGiven 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 $I_{a,b}(z,mu;m_1,m_2,l_1,l_2)$ and an $l_2$-dimensional hypergeometric integral $J_{a,b}(z,mu;m_1,m_2,l_1,l_2)$. The integrals depend on complex parameters $z$ and $mu$. We show that $I_{a,b}(z,mu;m_1,m_2,l_1,l_2)$ equals $J_{a,b}(e^mu,z;l_1,l_2,m_1,m_2)$ 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 $(gl_k,gl_n)$ duality for the qKZ and dynamical difference equations.
Source arXiv, math.QA/0309372
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