Jacobi Type Continued Fractions for the Ordinary Generating Functions of Generalized Factorial Functions

TL;DR

This paper introduces Jacobi type continued fractions for generalized factorial functions, extending known expansions and providing new generating functions, identities, and congruences for factorial-related sequences with broad combinatorial applications.

Contribution

It extends Flajolet's continued fraction expansions to generalized factorial functions, offering new formal power series approximations and applications in combinatorics.

Findings

Derived generalized J-fractions for factorial functions

Established new integer congruences for factorial-related sequences

Provided applications to combinatorial identities and harmonic numbers

Abstract

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.