I am trying to solve for a continuous function $h(z)$ given the following equation: $$h(z)=\left\{\begin{array} &g(z)\text{ when z is non-integer}\\ g_i(z)\text{ when z is integer}\\ \end{array}\right.$$$$g(z)=e^{z}\frac{\sin(\pi z)}{\pi}\sum_{k=1}^{\infty}\frac{e^{-2k}g_i(2k)}{z-2k}$$ $g_i(n)$ is a zero terminating recursive function defined for integer values by:
$$g{_i}(n)=\left\{\begin{array} &0\text{ when $n=0$} \\ 0\text{ when n is odd}\\ g_i(\frac{n}{2})+1\text{ when $n$ non-zero and even}\\ \end{array}\right.$$