Characterizations of Δ-Volterra lattice: A symmetric orthogonal polynomials interpretation

In this paper we introduce the Delta-Volterra lattice which is interpreted in terms of symmetric orthogonal polynomials. It is shown that the measure of orthogonality associated with these systems of orthogonal polynomials evolves in t like (1 + x(2))(1-t)mu(x) where mu is a given positive Borel measure. Moreover, the Delta-Volterra lattice is related to the Delta-Toda lattice from Miura or Backlund transformations. The main ingredients are orthogonal polynomials which satisfy an Appell condition with respect to the forward difference operator Delta and the characterization of the point spectrum of a Jacobian operator that satisfies a Delta-Volterra equation (Lax type theorem). We also provide an explicit example of solutions of Delta-Volterra and Delta-Toda lattices, and connect this example with the results presented in the paper. (C) 2015 Elsevier Inc. All rights reserved.