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Consequences of the Darboux transformations for the finite-difference Schrodinger equations and three-term recurrence relations for orthogonal polyno- mials are considered. An equivalence of the chain of these transformations, or discrete dressing chain, to the discrete-time Toda lattice is established. A more fundamental discrete-time Volterra lattice consisting of one simple difference- difference nonlinear equation is found. Some simple similarity reductions of these lattices are described. A group-theoretical meaning of the Darboux transforma- tions is illustrated on the example of Meixner polynomials. The general set of Askey-Wilson potentials is shown to define a class of solutions of the derived discrete-time equations. A subset of the latter potentials that can be obtained by dressing of the free discrete Schrodinger equation is characterized. A class of q-Racah polynomials for qN = 1 orthogonal with respect to a positive measure is obtained by undressing of the finite-dimensional Chebyshev polynomials.

Discrete Darboux transformations, the discrete-time Toda lattice, and the Askey–Wilson polynomials