Let $P(x)$ be a nonconstant integer polynomial with nonnegative coëfficiënts such that the equation $y= P(y)$ has only one real solution $q$.
Let $x_1=P(0)$ and $x_n = P(x_{n-1})$.
$$f(z) = \sum_{n>1} \frac{1}{x_n ^ z}$$
Consider the analytic continuation of $f(z) : F(z)$.
If $q$ is in the range of $F(r)$ for $dom(r)$ being the reals, and there are no poles for $F(A)$ where $A$ is the half-open interval $[-2,1[$ then
CONJECTURE
$$ F(-1) = q $$
Is this always true ?
What if there are 2 solutions $q_1,q_2$ ?