A general partial-fraction relation for derivatives of an entire function

Let $F$ be the function under consideration, with zeros $a_k$ indexed by $k\geq 0$, let $L$ be the derivative order, let $m$ be a positive integer, and let $\omega_m$ denote an $m$th root of unity. The notation $F^{(L)}(\infty)/F(\infty)$ denotes the corresponding value at infinity when it exists. General partial-fraction relation. The following more general relation is conjectured:

$$\frac{F^{(L)}(z)}{F(z)}=\frac{F^{(L)}(\infty)}{F(\infty)}+\sum_{k=0}^{\infty}\sum_{r=0}^{m-1}\frac{F^{(L)}(a_{k})}{F'(a_{k})}\left(\frac{\omega_{m}^{r}}{z-a_{k}\omega_{m}^{r}}\right).$$

The proposed identity is intended to generalize the preceding recursion formula for the Hurwitz zeta function and the partial-fraction decompositions developed in the paper; its validity and the required hypotheses on $F$, $L$, $m$, and convergence are not established in the supplied text.