Cross-post from MSE.

It is known that the sine can be expressed as an infinite product: $$\sin(x) = x \prod_{n=1}^{\infty} \Big{(} 1 - \frac{x^{2}}{n^{2}{\pi}^{2}} \Big{)} .$$ We can define that functional square root of a function $g(\cdot)$ to be the function $f(\cdot)$ that satisfies $f(f(x)) = g(x)$. The square root of the sine function with respect to function composition has been discussed previously on MO on a number of occasions. For instance, here the formal power series is considered.

I wonder whether the functional square root of the sine also has an infinite product representation. If not, has any research been done on this question?

thefunctional square root. Is it unique? $\endgroup$