Structure relations for orthogonal polynomials

Abstract

We show how to derive structure relations for general orthogonal polynomials, that is, we find operators whose action on pn is a combination of pn and pn+1 with variable coefficients. We also provide an analogue of the string equation for general orthogonal polynomials. We explore the connection with Toda lattice and polynomials orthogonal with respect to general exponential weights.