New series representations for any positive power of $\pi$ from a relation involving trigonometric functions

Abstract

In previous works, we presented series representations for $\pi^3$ and $\pi^5$, in which the prefactor depends only on the golden ratio appears. In this article, we derive a general relation involving trigonometric functions and an infinite series. Such an identity is likely to provide many series representations for any positive power of $\pi$, among them the above mentioned representations for $\pi^3$ and $\pi^5$.