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

TL;DR

This paper derives new series and product representations for powers of pi using Fourier series and derivatives of a specific function, connecting classical infinite product formulas with novel series expansions.

Contribution

It introduces a unified formalism to derive series and infinite product representations of pi and its powers from Fourier analysis and derivatives.

Findings

New series representations for positive powers of pi.

Derivation of Euler-Wallis product from the formalism.

Infinite product representations of pi obtained systematically.

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 $\pi$.