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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.

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

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 $k>2$?

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.

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 here on MO.)

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.

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