Laguerre-Freud's equations for the recurrence coefficients of semi-classical orthogonal polynomials

Abstract In this paper we give a new scheme for deriving a non-linear system, satisfied by the three-term recurrence coefficients of semi-classical orthogonal Polynomials, this non-linear system is labelled "Laguerre-Freud′s equations." Here we do not deal with the numerical aspect of the question (stability, asymptotic, ...). Our purpose is to take due advantage of linear functionals formalism and to show that given a semi-classical linear functional, i.e., given two polynomials, we are able to provide the Laguerre-Freud"s equations. The way of obtaining these equations is put in a recursive form appropriate for computer algebra calculation, especially when the degrees of the two given polynomials are large. We illustrate our process by several examples and we point out two cases where the solutions to Laguerre-Freud′s equations are not unique.