Orthogonal polynomial interpretation of Delta-Toda equations

Abstract

The correspondence between dynamics of Delta-Toda equations for the coefficients of the Jacobi operator and its resolvent function is established. A method to solve inverse problem - integration of Delta-Toda equations - based on Padé approximates and continued fractions for the resolvent function is proposed. The main ingredient are orthogonal polynomials which satisfy an Appell condition, with respect to the forward difference operator Delta. Two examples related with Jacobi and Laguerre orthogonal polynomials and Delta-Toda equations are given.