Combinatorial Identities for Generalized Stirling Numbers Expanding $f$-Factorial Functions and the $f$-Harmonic Numbers

TL;DR

This paper introduces a broad class of $f(t)$-factorials and explores their associated generalized Stirling numbers, revealing new combinatorial identities and extending classical harmonic number relations.

Contribution

It develops a unified framework for $f(t)$-factorials and their Stirling numbers, generalizing known factorial and harmonic number identities with new combinatorial properties.

Findings

Derived combinatorial identities for generalized Stirling numbers

Extended harmonic number relations to $f$-harmonic numbers

Unified framework encompassing many factorial variants

Abstract

We introduce a class of $f(t)$-factorials, or $f(t)$-Pochhammer symbols, that includes many, if not most, well-known factorial and multiple factorial function variants as special cases. We consider the combinatorial properties of the corresponding generalized classes of Stirling numbers of the first kind which arise as the coefficients of the symbolic polynomial expansions of these $f$-factorial functions. The combinatorial properties of these more general parameterized Stirling number triangles we prove within the article include analogs to known expansions of the ordinary Stirling numbers by $p$-order harmonic number sequences through the definition of a corresponding class of $p$-order $f$-harmonic numbers.