Laguerre–Freud equations associated with the D-Laguerre–Hahn forms of class one

Abstract

AbstractWe give the system of Laguerre–Freud equations for the recurrence coefficients{\beta_{n}},{\gamma_{n+1}},{n\geq 0}, of orthogonal polynomials with respect to a D-Laguerre–Hahn form (linear functional) of class one. The system is solved in the case when{\beta_{0}=-\theta_{0}},{\beta_{n+1}=\theta_{n}-\theta_{n+1}}and{\gamma_{n+1}=-\theta_{n}^{2}}with{\theta_{n}\neq 0},{n\geq 0}. There are essentially three canonical cases.