On a Curious q -Hypergeometric Identity

In this paper, we examine the limiting behavior of solutions to an infinite set of recursions involving q-factorial terms as q→1. The underlying problem is sensitive to small perturbations and the very existence of a limit, to say nothing of its precise form, is surprising. We determine it by showing that the task at hand is equivalent to the convergence of one set of orthogonal polynomials on the unit circle to another such set, Geronimus polynomials, as q→1.