Title: The Generalized Superfactorial, Hyperfactorial and Primorial functions Show Chinese title

Title: 广义超阶乘、超阶乘和素数阶乘函数

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: $n\$$). Show Chinese abstract

Abstract: 本文介绍了一种新的广义超阶乘函数（可称为$n^{th}$次超阶乘：$sf^{(n)}(x)$）和一种广义高阶乘函数（可称为$n^{th}$次高阶乘：$H^{(n)}(x)$），并证明这些函数具有涉及图数的显式公式。 除了讨论其他数列模式外，我们还介绍了一种广义素数阶乘函数和两个相关的定理。 请注意，Sloane 和 Plouffe（1995）提供的超阶乘定义是所考虑的定义（而不是 Clifford Pickover（1995）的超阶乘函数：$n\$$）。