Hey everyone,

I would like to know if anybody could help me find references for the following.

Take a suitably well defined entire function $f(x)$ and it's derivative $\tilde{f}(x)$ to which the roots $x_n$ and $\tilde{x}_n$ are associated. The function $f(x)$ may have infinitely many roots, though naturally one needs to be more careful in this case. Let's discount these subtleties for the time being. Define

$$Z(s)=\sum_n\frac{1}{x_n^s} \hspace{8mm} \textrm{and} \hspace{8mm} \tilde{Z}(s)=\sum_n\frac{1}{\tilde{x}_n^s}.$$

One of the identities which I have proven which relates these is

$$\tilde{Z}(3)=Z(3)-\left(\frac{Z(2)}{Z(1)}\right)^3+3\left(\frac{Z(3)Z(2)}{(Z(1))^2}-\frac{Z(4)}{Z(1)}\right).$$

I also have proven other identities (some much more simple) of this form. I have searched the internet, journals and every analysis book in my University library and found nothing of the sort. Also the main applications which I would expect to find curiously also do not appear in any of the aforementioned sources.

*Edit*

Just to say thanks to those who replied. The question has been answered :)