The Generalized Superfactorial, Hyperfactorial and Primorial functions

TL;DR

This paper introduces new generalized superfactorial, hyperfactorial, and primorial functions with explicit formulas involving figurate numbers, expanding the understanding of these mathematical constructs and their interrelations.

Contribution

It presents novel definitions for n-th degree superfactorial and hyperfactorial functions, along with a generalized primorial, supported by explicit formulas and new theorems.

Findings

Explicit formulas involving figurate numbers for the new functions

Introduction of a generalized primorial function and related theorems

Enhanced understanding of number patterns and functions

Abstract

This paper introduces a new generalized superfactorial function (referable to as $n^{th}$- degree superfactorial: $sf^{(n)}(x)$) and a generalized hyperfactorial function (referable to as $n^{th}$- degree hyperfactorial: $H^{(n)}(x)$), and we show that these functions possess explicit formulae involving figurate numbers. Besides discussing additional number patterns, we also introduce a generalized primorial function and 2 related theorems. Note that the superfactorial definition offered by Sloane and Plouffe (1995) is the definition considered (and not Clifford Pickover's (1995) superfactorial function: $n\$$).