Series representations for $\pi^3$ involving the golden ratio

TL;DR

This paper derives new series representations for ^3 involving the golden ratio, expanding the mathematical understanding of ^3 through trigonometric identities and infinite sums.

Contribution

It introduces novel series representations for ^3 using Euler's trigonometric identity and the golden ratio, a contribution not previously documented.

Findings

Two new series representations for ^3 involving the golden ratio

Methodology based on Euler's trigonometric identity

Potential to generalize to other constants

Abstract

Although many series exist for $\pi$ and $\pi^2$, very few are known for $\pi^3$. In this article, we derive, using a trigonometric identity obtained by Euler, two representations of $\pi^3$ involving infinite sums and the golden ratio. The methodology can be generalized in order to obtain further series, relating by the way $\pi^3$ to other mathematical constants.