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 power representation. If not, has any research been done on this question?

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?