An identity for the infinite sum $\sum_{n=0}^\infty\frac{1}{(n!)^3}$

TL;DR

This paper derives a new identity for the special case of the alpha function involving the sum of reciprocals of factorials cubed, expanding understanding of this mathematical series.

Contribution

It provides a novel explicit identity for the alpha function when s=3, which was previously not well-characterized.

Findings

Derived an explicit identity for the sum with s=3

Extended the understanding of the alpha function for specific parameters

Contributes to mathematical series and special functions literature

Abstract

In this short note, we give an identity for the $\alpha$ function $$\alpha(x,s)=\sum_{n=0}^\infty\frac{x^n}{(n!)^s}$$ where $s\in \mathbb{N}$, $x\in \mathbb{R}$, in the case $s=3$.