# Relationships between the roots of an entire function and the roots of its derivative

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 :)

• Sorry I have just realised that I completely misnamed the thread! Dec 9, 2010 at 19:33
• You can rename it if you edit the question. Dec 9, 2010 at 19:53

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 $a_j$.

• Right. The Z(s) here are the power symmetric functions, which are thoroughly studied in the literature. The \tilde{Z}(s) I am unsure about, but identities for the power symmetric functions will apply to them. Dec 9, 2010 at 19:46
• Hey, Yes the results do specialise to give the results which you provide. However they also cover general entire functions, providing they obey certain qualities. Do you have any idea where I might find a list of references for the above polynomial identities? It might help Thanks very much! - Joe Dec 9, 2010 at 19:46
• @Joe: symmetric function identities are covered in Stanley's Enumerative Combinatorics (I can't remember which part at the moment), and they are also covered in Sagan's The Symmetric Group. Dec 9, 2010 at 19:54
• @Qiaochu: Thank you, I found a copy on google books and I will have a read through. I guessed that my identities, when applied to polynomials, would surely be known. However it's seeming less likely that the results are known when applied to general entire functions Dec 9, 2010 at 19:58
• And also in Macdonald's Symmetric Functions and Hall Polynomials. I don't know off-hand if they have results like this though. Dec 9, 2010 at 19:59

Let $f(x) = (1 - r_1 x)...(1 - r_n x)$ be a polynomial. Then $f(x) = 1 - e_1 x^1 + e_2 x^2 \mp ...$ where the $e_i$ are the elementary symmetric functions in the $r_i$. We define also $p_k = \sum_i r_i^k$, the power symmetric functions in the $r_i$. Then Newton's identities state that

$$ke_k = \sum_{i=1}^{k} (-1)^{i-1} e_{k-i} p_i.$$

This identity is equivalent to the generating function identity

$$\frac{f'(x)}{f(x)} = \sum_{i=1}^{n} \frac{r_i}{1 - r_i x} = \sum_{k \ge 0} p_{k+1} x^k$$

which follows from taking logarithmic derivatives on both sides. Now, you want to relate the functions $p_k$ to the functions $\tilde{p}_k$, the power symmetric functions of the reciprocals of the roots of the derivative $f'(x)$. Applying Newton's identities to $f'(x) = -e_1 + 2e_2 x - 3e_3 x^2 \pm ...$ gives

$$k(k+1) \frac{e_{k+1}}{e_1} = \sum_{i=1}^k (-1)^{i-1} \frac{(k+1-i) e_{k+1-i}}{e_1} \tilde{p}_i.$$

This pair of identities should get you your results (at least formally, letting $n \to \infty$). You may or may not find it useful to write Newton's identities in the equivalent form

$$e_n = \frac{1}{n!} \sum_{\sigma \in S_n} \text{sgn}(\sigma) p_{\sigma}$$

where $p_{\sigma} = p_{\lambda_1} ... p_{\lambda_i}$, where $\sigma$ has cycle type $(\lambda_1, ... \lambda_i)$. This gives

$$\frac{(n+1) e_{n+1}}{e_1} = \frac{1}{n!} \sum_{\sigma \in S_n} \text{sgn}(\sigma) \tilde{p}_{\sigma}.$$

• Thank you again for your reply. This method seems similar to my method and I can see why it would give the same results, at least in the finite case :) Dec 9, 2010 at 20:24
• My method uses repeated derivatives of $\frac{f'(x)}{f(x)}$ evaluated at $x=0$ on infinite products. This is rigorous providing that the roots of the product grow at a sufficient rate for the product (in Hadamard form) to converge. Dec 9, 2010 at 20:29
• @Joe: the passage from the finite to the infinite case should be straightforward. Dec 9, 2010 at 20:33
• @ Qiaochu: It is not completely straightforward as there is no assurance (that I can find) that the sum of the roots of $f'(x)$ should converge in the infinite case. In fact I only have on concrete example where I can prove that this is the case Dec 9, 2010 at 20:59