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

Abstract

Consequences of the Darboux transformations for the finite-difference Schrodinger equations and three-term recurrence relations for orthogonal polynomials 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 differencedifference nonlinear equation is found. Some simple similarity reductions of these lattices are described. A group-theoretical meaning of the Darboux transformations 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 g-Racah polynomials for q N = 1 orthogonal with respect to a positive measure is obtained by undressing of the finite-dimensional Chebyshev polynomials.