Question

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?

Answer 1

(Too long for a comment.)

It's a bit older than your reference, but so-called "hypergoniometric functions" have been considered by Erik Lundberg in 1879. This article is a more recent discussion. Shelupsky and Burgoyne discuss similar generalizations. All ultimately consider this as the problem of inverting an appropriate generalization of the integral representations of arcsine and arccosine.

The $n=3$ case has been considered separately by A.C. Dixon; I had talked a bit about Dixon elliptic functions in this math.SE answer.