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The Cholesky factorization of the moment matrix is considered for thegeneralized Charlier, generalized Meixner, and Gauss hypergeometric discreteorthogonal polynomials.  For the generalized Charlier, we present an alternative derivation of theLaguerre-Freud relations found by Smet and Van Assche. Third-order andsecond-order nonlinear ordinary differential equations are found for therecursion coefficient $\gamma_n$, that happen to be forms of the Painlev\'e$\text{deg-P}_{\text V}$ in disguise. Laguerre-Freud relations are also foundfor the generalized Meixner case, which are compared with those of Smet and VanAssche.  Finally, the Gauss hypergeometric discrete orthogonal polynomials, also knownas generalized Hahn of type I, are also studied. Laguerre-Freud equations arefound, and the differences with the equations found by Dominici and by Filipukand Van Assche are provided.

Laguerre-Freud Equations for the Generalized Charlier, Generalized Meixner and Gauss Hypergeometric Orthogonal Polynomials