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:


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



Must $q$ be analytic?
No. $F$ does not put enough constraints on $q$.


Suppose that $F$ is a given analytic function, and let $q\_0:[0, 1]\to \mathbb R\_+$ be continuous. Extend $q\_0$ to a function on $\mathbb R\_+$ using the recurrence
$$q(s+1) = (e^{q(s)}-e^s)F(s).$$
In general, $q$ will not be continuous or positive. One an write down conditions for that, but they are not needed to answer this question.


Let $q$ be one function that satisfies all assumptions. Let $q'\_0=q|\_{(0,1]} + g$, where $g$ is a non-negative continuous function on $[0,1]$ with $g'(0)=g'(1)=0$, CBS extend $q'\_0$ to a function on $\mathbb R\_+$ as sketched above. If $q$ is analytic but $g$ not, then $q'$ will not be analytic either.


