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

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 \(t_{n}\ne 0n\ge 0\), where \(D_w\) is the divided difference operator. There are essentially four canonical families.