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