From Fourier series to infinite product representations of $\pi$ and infinite-series forms for its positive powers

Abstract

In this article, we derive, using Fourier series and multiple derivative of the function $\pi /\sin (\pi x)$, series representations for positive powers of $\pi$. We also show that the Euler-Wallis product can be easily obtained from the same formalism and deduce infinite products representations of $\pi$.