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Cross-post from MSE.

The Jacobi-Anger expansion 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. Moreover, we have $\sin^{[3]}(\theta) = \sin(\sin(\sin(\theta))) $, and $\sin^{[k]}(\theta)$ is the function in which there are $k-1$ compositions of the sine function with itself (when $k\geq2$).

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

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    $\begingroup$ Isn't this just the Fourier expansion of odd periodic functions such as $\,\sin^{[k]}(\theta)\,$ whose coefficients are certain integrals? $\endgroup$
    – Somos
    Commented Nov 22, 2020 at 20:00
  • $\begingroup$ @Somos Yes, I think so. At least for $k=2$, the two expressions seem to coincide: math.stackexchange.com/questions/877408/fourier-series-sin-sinx $\endgroup$
    – Max Lonysa Muller
    Commented Nov 22, 2020 at 23:08
  • $\begingroup$ here is one representation: math.stackexchange.com/q/4297398/87355 $\endgroup$
    – Carlo Beenakker
    Commented Dec 18, 2021 at 23:02
  • $\begingroup$ @CarloBeenakker interesting, thank you. Would be great if the Bessel sum coefficients could be simplified and written in a closed form somehow $\endgroup$
    – Max Lonysa Muller
    Commented Dec 19, 2021 at 15:36

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