Generalized trigonometric functions $Cos(n) v$ and $Sin(n) v$.

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.

Answer 2

The existing literature on generalized trigonometric functions is scarce and it seems there isn't a comprehensive account of generalized trigonometric functions anywhere. Also there isn't a unified accepted notation and different authors use different notations. The generalized sine and cosine functions $\sin_{pr}x$, $\ \cos_{pr}x$ are defined by the formulas $$ x=\int_0^{\sin_{pr}x}\frac{dt}{\sqrt[p]{1-t^r}},\qquad \cos_{pr}x=\sqrt[r]{1-(\sin_{pr}x)^r}. $$For generic values of parameters $p,r$ it appears that these functions are not very interesting, but when these parameters have special values, then $\sin_{pr}x$ can be expressed algebraically through Jacobi elliptic functions, and as a consequence one can establish addition theorems analogous to addition theorems for elliptic functions.

Below I give a unified relatively simple discussion of several such cases. The first case $r=2,\ p=2$ is trivial. The second case $r=3,\ p=\frac{3}{2}$ is due to Cayley and Dixon. The third case $r=4,\ p=\frac{4}{3}$ has been considered relatively recently by Edmunds, Gurka, and Lang (Properties of generalized trigonometric functions, J. Approx. Theory 164 (2012), no. 1, 47-56.) I couldn't find the discussion of the fourth case $r=6,\ p=\frac{6}{5}$ in the literature, but it easily follows from the same considerations as in the first three cases. The parameter $r$ is denoted in analogy with Ramanujan's theory of elliptic functions to alternative bases (see "Ramanujan's notebooks, vol. 5" by Bruce Berndt). I don't know whether it is just a coincidence or there is a deeper connection, but $r=2,3,4,6$ are the four signatures for which such alternative theories of elliptic functions have been developed.

Let's consider the case when $\frac{1}{p}=1-\frac{1}{r}$. Then by a series of simple changes of variables one obtains \begin{align} \int_0^{u}\frac{dt}{(1-t^r)^{1-1/r}}=\frac{1}{r}\int_0^{u^r}\frac{dt}{(1-t)^{1-1/r}t^{1-1/r}}=\tag{1}\\ \frac{1}{r}\int_0^{u^r}\frac{dt}{(t-t^2)^{1-1/r}}=\frac{1}{r}\int_0^{u^r}\frac{dt}{\left(\frac{1}{4}-\left(\frac{1}{2}-t\right)^2\right)^{1-1/r}}=\\ \frac{2^{2-2/r}}{r}\int\limits_{1-2u^r}^{1}\frac{dt}{(1-t^2)^{1-1/r}}=\frac{2^{1-2/r}}{r}\int\limits_{(1-2u^r)^2}^{1}\frac{dt}{\sqrt{t}(1-t)^{1-1/r}}=\\ \frac{2^{1-2/r}}{r}\int\limits_0^{1-(1-2u^r)^2}\frac{t^{1/r-1}dt}{\sqrt{1-t}}=2^{1-2/r}\cdot\int\limits_0^\sqrt[r]{1-(1-2u^r)^2}\frac{dt}{\sqrt{1-t^r}} \end{align} When $r=2,3,4$ the last integral can be inverted in terms of elliptic functions. But the case $r=6$ also can be inverted, since $$ \int\frac{dt}{\sqrt{1-t^6}}=-\frac{1}{2}\int\frac{d\left(\frac{1}{t^2}\right)}{\sqrt{\frac{1}{t^6}-1}}. $$

As an illustration, according to Edmunds,Gurka, and Lang ($r=4$) one has $$ \sqrt{1-\sqrt{\frac{1-\left(\sin_{\frac{4}{3},4}x\right)^2}{1+\left(\sin_{\frac{4}{3},4}x\right)^2}}}=\text{sn}\left(x,\frac{1}{\sqrt{2}}\right). $$

Representation in terms of elliptic functions is possible also when $\frac{1}{p}\neq 1-\frac{1}{r}$. This can be seen by substitution $t\to \frac{at+b}{ct+d}$ in the integral (second integral in eq.$(1)$) $$ \int\frac{dt}{(1-t)^{1-1/r}t^{1-1/r}}, \quad (r=2,3,4,6), $$ and specifying parameters $a,b,c,d$ such that the resulting integral again has he form of an incomplete beta function.