I just discovered a paper from 1948, *Eine Verallgemeinerung der Kreis-und Hyperbelfunktionen* by R. Grammel which introduces functions he calls Cos(n) and Sin(n), representing a parameterization of the curve $x^n + y^n=1$ in $\mathbb{R}^2$ (the unit sphere of the n-norm) (also, I know the notation is pretty bad, if it was my choice I'd probably write something like $\sin_n$).

Grammel then proceeds to prove many identities about these generalizations of the circular sine and cosine that seem to show that they have much in common with the usual trigonometric functions.

Trying to find more information about these functions, I did not succeed in finding anything recent. I wondered if that was perhaps only because the terminology has changed since that paper or if there was some modern sense in which the study of these functions is trivial or uninteresting?