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 :)



Relationships between the roots of an entire function and the roots of its derivative
I don't know if there are any general results about these, but
when $f$ is a polynomial, these must be in essence results about
symmetric functions. If $f(z)=z^n+a\_{n-1} z^{n-1}+\cdots+a\_1z+a\_0$
then $Z(1)=-a\_1/a\_0$, $Z(2)=Z(1)^2-2a\_2/a\_0$ etc. In this case
your results surely specialize to polynomial identities in the $a\_j$.


