Basic trigonometric power sums with applications

TL;DR

This paper transforms sums of sine and cosine powers into combinatorial forms and applies these results to generating functions and counting closed walks on paths and cycles.

Contribution

It introduces a method to convert trigonometric power sums into combinatorial expressions and applies it to problems in graph theory and generating functions.

Findings

Derived combinatorial formulas for trigonometric power sums

Connected these formulas to generating functions

Counted closed walks on paths and cycles

Abstract

We present the transformation of several sums of positive integer powers of the sine and cosine into non-trigonometric combinatorial forms. The results are applied to the derivation of generating functions and to the number of the closed walks on a path and in a cycle.