Laguerre-Freud equations for the recurrence coefficients of \(D_{\omega}\) semi-classical orthogonal polynomials of class one (Q1298557)

The authors consider orthogonal polynomials \(P_n(x)\) with respect to a \(D_\omega\) semi-classical linear form, where \(D_\omega\) denotes the difference operator \[ D_\omega P(x)={{P(x+\omega)-P(x)}\over{\omega}}, \qquad D_1=\Delta, D_{-1}=\nabla. \] They derive the Laguerre-Freud equations which yield the recurrence coefficients for \(P_n(x)\). Moreover they consider the cases \(\omega=1\) as well as the limit \(\omega\rightarrow 0\) to compare their results with known ones in the classical discrete and continuous cases. Finally the generalized Meixner case is studied in detail.