Trigonometric functions in the $p$-norm

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 $\ddot{I}€$ values.

16 pages, 6 figures, accepted for publication in Mathematics Exchange