Identities between q-hypergeometric and hypergeometric integrals of different dimensions

Given complex numbers m1,l1 and nonnegative integers m2,l2, such that m1+m2=l1+l2, for any a,b=0,…,min(m2,l2) we define an l2-dimensional Barnes type q-hypergeometric integral Ia,b(z,μ;m1,m2,l1,l2) and an l2-dimensional hypergeometric integral Ja,b(z,μ;m1,m2,l1,l2). The integrals depend on complex parameters z and μ. We show that Ia,b(z,μ;m1,m2,l1,l2) equals Ja,b(eμ,z;l1,l2,m1,m2) up to an explicit factor, thus establishing an equality of l2-dimensional q-hypergeometric and m2-dimensional hypergeometric integrals. The identity is based on the (glk,gln) duality for the qKZ and dynamical difference equations.