A function with $R_0(f)=\sum_{n\ge0} \frac{1}{n!(n+1)!}$

Question

I'm looking for a function $f$ that satisfies the following: $$R_0(f) = \sum_{n\ge0} \frac{1}{n!(n+1)!}$$ and $f$ is holomorphic on $\mathbb{C}/\{0\}$.
I was thinking about something like $$\frac{e^z}{log(z)}$$ but it doesn't work. Has anyone an idea?

Answer 1

If $f(z)= \frac cz$, then $f$ is holomorphic on $\mathbb{C}/\{0\}$ and $R_0(f)=c$. Hence , with $c= \sum_{n\ge0} \frac{1}{n!(n+1)!}$, the function $f$ has the desired properties.

Answer 2

Note: Your series is a Bessel function value... $$ \sum_{n=0}^\infty \frac{1}{n!(n+1)!} = I_1(2) $$ where $I_1(x)$ is the modified Bessel Function of the first kind, a solution of the differential equation $$ x^2 y'' + xy' + (x^2-1)y = 0 $$ with Taylor series $$ \sum_{n=0}^\infty \frac{1}{n!(n+1)!}\left(\frac{x}{2}\right)^{2n+1} $$