A combinatorial approach to the Fourier expansions of powers of cos and sin

TL;DR

This paper introduces a novel combinatorial method for deriving Fourier expansions of powers of sine and cosine functions, and applies it to compute expansions of related rational functions.

Contribution

It provides a new combinatorial approach to Fourier series of powers of sine and cosine, extending to specific rational functions.

Findings

Derived Fourier expansions for powers of cos and sin

Computed Fourier series for 1/(a-cos t) and 1/(a-sin t)

Introduced a combinatorial framework for Fourier analysis

Abstract

We present a new combinatorial approach to the computation of the (real) Fourier expansions of $\cos^n(t)$ and $\sin^n(t)$, where $n\geq 1$ is an integer. As an application, we compute the Fourier expansions of $f(t)=\frac{1}{a-\cos t}$ and $g(t)=\frac{1}{a-\sin t}$, where $a\in\mathbb R$ with $|a|>1$.