Identities between q-hypergeometric and hypergeometric integrals of different dimensions

Details Identities between q-hypergeometric and hypergeometric integrals of different dimensions Identities between q-hypergeometric and hypergeometric integrals of different dimensions

2003-09-23

arxiv.org/abs/math/0309372v3

Mathematics

Quantum Algebra

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

Scientific paper

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 $I_{a,b}(z,\mu;m_1,m_2,l_1,l_2)$ 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.

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