\BOOKMARK [1][-]{section.1}{Notation and other conventions in the article}{}% 1
\BOOKMARK [2][-]{subsection.1.1}{Notation and special sequences}{section.1}% 2
\BOOKMARK [2][-]{subsection.1.2}{Mathematica summary notebook document and computational reference information}{section.1}% 3
\BOOKMARK [1][-]{section.2}{Introduction}{}% 4
\BOOKMARK [2][-]{subsection.2.1}{Polynomial expansions of generalized -factorial functions}{section.2}% 5
\BOOKMARK [2][-]{subsection.2.2}{Divergent ordinary generating functions approximated by the convergents to infinite Jacobi-type and Stieltjes-type continued fraction expansions}{section.2}% 6
\BOOKMARK [3][-]{subsubsection.2.2.1}{Infinite J-fraction expansions generating the rising factorial function}{subsection.2.2}% 7
\BOOKMARK [3][-]{subsubsection.2.2.2}{Examples}{subsection.2.2}% 8
\BOOKMARK [2][-]{subsection.2.3}{Generalized convergent functions generating factorial-related integer product sequences}{section.2}% 9
\BOOKMARK [3][-]{subsubsection.2.3.1}{Definitions of the generalized J-fraction expansions and the generalized convergent function series}{subsection.2.3}% 10
\BOOKMARK [3][-]{subsubsection.2.3.2}{Properties of the generalized J-fraction convergent functions}{subsection.2.3}% 11
\BOOKMARK [1][-]{section.3}{New results proved within the article}{}% 12
\BOOKMARK [2][-]{subsection.3.1}{A summary of the new results and outline of the article topics}{section.3}% 13
\BOOKMARK [2][-]{subsection.3.2}{Examples of factorial-related finite product sequences enumerated by the generalized convergent functions}{section.3}% 14
\BOOKMARK [3][-]{subsubsection.3.2.1}{Generating functions for arithmetic progressions of the -factorial functions}{subsection.3.2}% 15
\BOOKMARK [3][-]{subsubsection.3.2.2}{Generating functions for multi-valued integer product sequences}{subsection.3.2}% 16
\BOOKMARK [3][-]{subsubsection.3.2.3}{Examples of new convergent-based generating function identities for binomial coefficients}{subsection.3.2}% 17
\BOOKMARK [2][-]{subsection.3.3}{Examples of new congruences for the -factorial functions, the Stirling numbers of the first kind, and the r-order harmonic number sequences}{section.3}% 18
\BOOKMARK [3][-]{subsubsection.3.3.1}{Congruences for the -factorial functions modulo 2}{subsection.3.3}% 19
\BOOKMARK [3][-]{subsubsection.3.3.2}{New forms of congruences for the -factorial functions modulo 3, modulo 4, and modulo 5}{subsection.3.3}% 20
\BOOKMARK [3][-]{subsubsection.3.3.3}{New congruence properties for the Stirling numbers of the first kind}{subsection.3.3}% 21
\BOOKMARK [3][-]{subsubsection.3.3.4}{New congruences and rational generating functions for the r-order harmonic numbers}{subsection.3.3}% 22
\BOOKMARK [1][-]{section.4}{The Jacobi-type J-fractions for generalized factorial function sequences}{}% 23
\BOOKMARK [2][-]{subsection.4.1}{Enumerative properties of Jacobi-type J-fractions}{section.4}% 24
\BOOKMARK [2][-]{subsection.4.2}{A short direct proof of the J-fraction representations for the generalized product sequence generating functions}{section.4}% 25
\BOOKMARK [2][-]{subsection.4.3}{Alternate exact expansions of the generalized convergent functions}{section.4}% 26
\BOOKMARK [1][-]{section.5}{Properties of the generalized convergent functions}{}% 27
\BOOKMARK [2][-]{subsection.5.1}{The convergent denominator function sequences}{section.5}% 28
\BOOKMARK [2][-]{subsection.5.2}{The convergent numerator function sequences}{section.5}% 29
\BOOKMARK [3][-]{subsubsection.5.2.1}{Alternate forms of the convergent numerator function subsequences}{subsection.5.2}% 30
\BOOKMARK [1][-]{section.6}{Applications and motivating examples}{}% 31
\BOOKMARK [2][-]{subsection.6.1}{Lemmas}{section.6}% 32
\BOOKMARK [2][-]{subsection.6.2}{New congruences for the -factorial functions, the generalized Stirling number triangles, and Pochhammer k-symbols}{section.6}% 33
\BOOKMARK [3][-]{subsubsection.6.2.1}{Congruences for the Stirling numbers of the first kind and the r-order harmonic number sequences}{subsection.6.2}% 34
\BOOKMARK [3][-]{subsubsection.6.2.2}{Generalized expansions of the new integer congruences for the -factorial functions and the symbolic product sequences}{subsection.6.2}% 35
\BOOKMARK [2][-]{subsection.6.3}{Applications of rational diagonal-coefficient generating functions and Hadamard product sequences involving the generalized convergent functions}{section.6}% 36
\BOOKMARK [3][-]{subsubsection.6.3.1}{Generalized definitions and coefficient extraction formulas for sequences involving products of rational generating functions}{subsection.6.3}% 37
\BOOKMARK [3][-]{subsubsection.6.3.2}{Examples: Constructing hybrid rational generating function approximations from the convergent functions enumerating the generalized factorial product sequences}{subsection.6.3}% 38
\BOOKMARK [2][-]{subsection.6.4}{Examples: Expanding arithmetic progressions of the single factorial function}{section.6}% 39
\BOOKMARK [3][-]{subsubsection.6.4.1}{Expansions of arithmetic progression sequences involving the double factorial function \(a := 2\)}{subsection.6.4}% 40
\BOOKMARK [3][-]{subsubsection.6.4.2}{Expansions of arithmetic progression sequences involving the triple factorial function \(a := 3\)}{subsection.6.4}% 41
\BOOKMARK [3][-]{subsubsection.6.4.3}{Other special cases involving the quadruple and quintuple factorial functions \( a:= 4,5\)}{subsection.6.4}% 42
\BOOKMARK [2][-]{subsection.6.5}{Examples: Generalized superfactorial function products and relations to the Barnes G-function}{section.6}% 43
\BOOKMARK [3][-]{subsubsection.6.5.1}{Generating generalized superfactorial product sequences}{subsection.6.5}% 44
\BOOKMARK [3][-]{subsubsection.6.5.2}{Special cases of the generalized superfactorial products and their relations to the Barnes G-function at rational z}{subsection.6.5}% 45
\BOOKMARK [2][-]{subsection.6.6}{Examples: Enumerating sequences involving sums of factorial-related functions, sums of factorial powers, and more challenging combinatorial sums involving factorial functions}{section.6}% 46
\BOOKMARK [3][-]{subsubsection.6.6.1}{Generating sums of double and triple factorial powers}{subsection.6.6}% 47
\BOOKMARK [3][-]{subsubsection.6.6.2}{Another convergent-based generating function identity}{subsection.6.6}% 48
\BOOKMARK [3][-]{subsubsection.6.6.3}{Enumerating more challenging combinatorial sums involving double factorials}{subsection.6.6}% 49
\BOOKMARK [3][-]{subsubsection.6.6.4}{Other examples of convergent-based generating function identities enumerating the subfactorial function}{subsection.6.6}% 50
\BOOKMARK [2][-]{subsection.6.7}{Examples: Generating sums of powers of natural numbers, binomial coefficient sums, and sequences of binomials}{section.6}% 51
\BOOKMARK [3][-]{subsubsection.6.7.1}{Generating variants of sums of powers sequences}{subsection.6.7}% 52
\BOOKMARK [3][-]{subsubsection.6.7.2}{Semi-rational generating function constructions enumerating sequences of binomials}{subsection.6.7}% 53
\BOOKMARK [1][-]{section.7}{Conclusions}{}% 54
\BOOKMARK [1][-]{section.8}{Acknowledgments}{}% 55
\BOOKMARK [1][-]{section.9}{Appendix: Tables}{}% 56