Summary: The article studies a class of generalized factorial functions and symbolic product sequences through Jacobi-type continued fractions (J-fractions) that formally enumerate the typically divergent ordinary generating functions of these sequences. The rational convergents of these generalized J-fractions provide formal power series approximations to the ordinary generating functions that enumerate many specific classes of factorial-related integer product sequences. The article also provides applications to a number of specific factorial sum and product identities, new integer congruence relations satisfied by generalized factorial-related product sequences, the Stirling numbers of the first kind, and the $r$-order harmonic numbers, as well as new generating functions for the sequences of binomials, $m^{p} - 1$, among several other notable motivating examples given as applications of the new results proved in the article.

05A10, 05A15, 11A55, 11Y55, 11Y65, 11B65

continued fraction, J-fraction, S-fraction, Pochhammer symbol, factorial function, multifactorial, multiple factorial, double factorial, superfactorial, rising factorial, Pochhammer k-symbol, Barnes G-function, hyperfactorial, subfactorial, triple factorial, generalized Stirling number, Stirling number of the first kind, confluent hypergeometric function, la- guerre polynomial, ordinary generating function, diagonal generating function, Hadamard product, divergent ordinary generating function, formal Laplace-Borel transform, Stirling number congruence