  We use the rationality of the generalized $h^{th}$ convergent functions,
$Conv_h(\alpha, R; z)$, to the infinite J-fraction expansions enumerating the
generalized factorial product sequences, $p_n(\alpha, R) =
R(R+\alpha)\cdots(R+(n-1)\alpha)$, defined in the references to construct new
congruences and $h$-order finite difference equations for generalized factorial
functions modulo $h \alpha^t$ for any primes or odd integers $h \geq 2$ and
integers $0 \leq t \leq h$. Special cases of the results we consider within the
article include applications to new congruences and exact formulas for the
$\alpha$-factorial functions, $n!_{(\alpha)}$. Applications of the new results
we consider within the article include new finite sums for the
$\alpha$-factorial functions, restatements of classical necessary and
sufficient conditions of the primality of special integer subsequences and
tuples, and new finite sums for the single and double factorial functions
modulo integers $h \geq 2$.
