Trigonometric functions in the $p$-norm

TL;DR

This paper explores generalized trigonometric functions defined on the unit p-circle, revealing connections to various mathematical concepts such as double angle formulas, Stirling numbers, and gamma functions.

Contribution

It introduces and analyzes generalized trigonometric functions for the p-norm, extending classical trigonometry to non-Euclidean norms and uncovering new mathematical relationships.

Findings

Connections to double angle formulas

Relations with Stirling numbers and Bell polynomials

Generalized pi values for p-norms

Abstract

Trigonometry is the study of circular functions, which are functions defined on the unit circle $x^2+y^2 =1$, where distances are measured using the Euclidean norm. When distances are measured using the $L_p$-norm, we get generalized trigonometric functions. These are parametrizations of the unit $p$-circle $|x|^p+|y|^p =1$. Investigating these new functions leads to interesting connections involving double angle formulas, norms induced by inner products, Stirling numbers, Bell polynomials, Lagrange inversion, gamma functions, and generalized $\pi$ values.