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

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.