  The article studies a class of generalized factorial functions and symbolic
product sequences through Jacobi type continued fractions (J-fractions) that
formally enumerate the divergent ordinary generating functions of these
sequences. The more general definitions of these J-fractions extend the known
expansions of the continued fractions originally proved by Flajolet that
generate the rising factorial function, or Pochhammer symbol, $(x)_n$, at any
fixed non-zero indeterminate $x \in \mathbb{C}$. 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 identities,
new integer congruence relations satisfied by generalized factorial-related
product sequences and the $r$-order harmonic numbers, among several other
notable motivating examples as immediate applications of the new results. In
this sense, the article serves as a semi-comprehensive, detailed survey
reference that introduces applications to many established and otherwise
well-known combinatorial identities, new cases of generating functions for
factorial-function-related product sequences, and other examples of the
generalized integer-valued multifactorial, or $\alpha$-factorial, function
sequences. The convergent-based generating function techniques illustrated by
the particular examples cited within the article are easily extended to
enumerate the factorial-like product sequences arising in the context of many
other specific applications.
