New Congruences and Finite Difference Equations for Generalized Factorial Functions

Abstract: 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 $h \geq 2$.