<a href="https://math.stackexchange.com/questions/3910272/what-is-the-jacobi-anger-expansion-of-the-kth-functional-iterate-of-the-sine">Cross-post</a> from MSE.

The Jacobi-Anger <a href="https://en.wikipedia.org/wiki/Jacobi%E2%80%93Anger_expansion">expansion</a> gives expressions for functional iterates of trigonometric functions in terms of Bessel functions. For instance, we have $$\sin(z \sin(\theta)) = 2 \sum_{n=1}^{\infty} J_{2n-1} (z) \sin[(2n-1)\theta] .$$
If we plug in $z=1$, we obtain an expression for $\sin(\sin(\theta)) := \sin^{[2]}(\theta)$. We might call this the second functional iterate of the sine function. 

**Question**: Are Jacobi-Anger expansions for the $k$'th functional iterate -- $\sin^{[k]}(\theta)$ -- also known, for arbitrary $k>2$?

(My motivation for this question mainly stems from the expression for the functional square root of a function, as seen over <a href="https://mathoverflow.net/questions/17605/how-to-solve-ffx-cosx/44727#44727">here</a> on MO.)