The addition formulae for the Jacobi elliptic functions are given by
$sn(u+v)=\frac{sn(u)cn(v)dn(v)+cn(u)sn(v)dn(v)}{1-k^2sn^2(u)sn^2(v)}$,
$cn(u+v)=\frac{cn(u)cn(v)-sn(u)dn(u)sn(v)dn(v)}{1-k^2sn^2(u)sn^2(v)}$,
and
$dn(u+v)=\frac{dn(u)dn(v)-k^2sn(u)cn(u)sn(v)cn(v)}{1-k^2sn^2(u)sn^2(v)}$.
Using these it is easy to compute what $sn(2u)$, $cn(2u)$ and $dn(2u)$ are in a relatively simple form. I'm interested in whether there are formulae in terms of the original functions $sn(u)$ etc for multiples such as $sn(nu)$ etc for $n\in\mathbb{Z}$? I tried doing this iteratively just for $sn(3u)$ and it doesn't simplify obviously.
Ideally, what I'm looking for is something analogous to the trigonometric multiple angle identities:
$\sin(n\theta)=\sum_{r=0}^n{\begin{pmatrix}n\\r\end{pmatrix}\cos^r(\theta)\sin^{n-r}(\theta)\sin(\frac{1}{2}(n-r)\pi)}$,
$\cos(n\theta)=\sum_{r=0}^n{\begin{pmatrix}n\\r\end{pmatrix}\cos^r(\theta)\sin^{n-r}(\theta)\cos(\frac{1}{2}(n-r)\pi)}$,
and
$\tan(n\theta)=\frac{\tan((n-1)\theta)+\tan(\theta)}{1-\tan((n-1)\theta)\tan(\theta)}$,
but I can't seem to find anything anywhere like this. Even if the formulae for general $n$ are not known, I'm mostly interested in the case $n=3$, so any insight would be appreciated greatly!