A family of discrete semi-classical orthogonal polynomials of class one

Abstract

In this work, we solve the system of Laguerre–Freud equations for the recurrence coefficients $\beta _n$, $\gamma _{n+1} , n \ge 0$ of the $D_{w}$-semi-classical orthogonal polynomials sequences of class one in the case when $\beta _{0}=-t_{0}$, $\beta _{n+1}=t_{n}-t_{n+1}$ and $\gamma _{n+1}=-t_{n}^{2}$ with $t_{n}\ne 0\;n\ge 0$, where $D_w$ is the divided difference operator. There are essentially four canonical families.