Compute $f(x)$ given $\sum_{n=0}^\infty f((n+\frac{1}{2})z)$

Question

Let's say I have some function $$F(z) = \sum_{n=0}^\infty f\big(\big(n+\tfrac{1}{2}\big)z\big)$$ which I can evaluate for any $z>0$. Is there a way to recover $f$ purely by evaluations of $F$? I'm thinking there might be some inversion method similar to the Ramanujan interpolation formula, but I'm unsure how to proceed.

Answer 1

$F(z) = \sum_{n \geq 0} f((n+0.5)z)$

Let $f(z) = \sum_{m \leq L} a_m z^m$ then

$$F(z) = \sum_{n \geq 0} \sum_{m \leq L} a_m ((n+0.5)z)^m = \sum_{n \geq 0} \sum_{m \leq L} a_m (n+0.5)^m z^m = \sum_{m \leq L} b_m z^m$$

We can find: $$\implies b_m = a_m \sum_{n \geq 0}(n+0.5)^m .$$ for $m \leq L = -2$.

But similarly solving assuming series expansion for $f(z)$ at $z=a$ is involved.