On generalized trigonometric functions and series of rational functions

TL;DR

This paper introduces a framework for constructing generalized trigonometric functions linked to complex polynomials, extending classical identities and enabling new evaluations of infinite rational series.

Contribution

It presents a novel method to define generalized trigonometric functions for any complex polynomial, generalizing classical identities and applying them to evaluate infinite series.

Findings

Generalized trigonometric functions satisfy algebraic identities extending Pythagorean theorem.

These functions can be used to evaluate infinite series of rational functions.

Classical sine and cosine are special cases associated with quadratic polynomial x^2-1.

Abstract

Here we introduce a way to construct generalized trigonometric functions associated with any complex polynomials, and the well known trigonometric functions can be seen to associate with polynomial $x^2-1$. We will show that those generalized trigonometric functions have algebraic identities which generalizes the well known $\sin^2(x)+\cos^2(x)=1$. One application of the generalized trigonometric functions is evaluating infinite series of rational functions.