I have a continuous function $q:\mathbb{R}^+ \to \mathbb{R}^+$. An interesting property of this function is that

$$F(s) = \frac{e^{-q(s)}q(s+1)}{1-e^{s-q(s)}}$$

which also takes $\mathbb{R}^+ \to \mathbb{R}^+$, can be analytically continued to an entire function on $\mathbb{C}$.

Because of this, I wonder:

Must $q$ be analytic on $\mathbb{R}^+$?

If it helps, $F(s)$ satisfies the flippant equation:

$$F(s) = e^{F(s-1)e^{s-1}}$$