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We consider orthogonal polynomials on the unit circle with respect to aweight which is a quotient of $q$-gamma functions. We show that the Verblunskycoefficients of these polynomials satisfy discrete Painlev\'e equations, in aLax form, which correspond to an $A_3^{(1)}$ surface in Sakai's classification.

Orthogonal polynomials on the unit circle, $q$-Gamma weights, and discrete Painlev\'e equations