# 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 :)

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Sorry I have just realised that I completely misnamed the thread! –  backstoreality Dec 9 '10 at 19:33
You can rename it if you edit the question. –  Ryan Reich Dec 9 '10 at 19:53
Thank you Ryan :) –  backstoreality Dec 9 '10 at 20:12

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 $a_j$.

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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. –  Qiaochu Yuan Dec 9 '10 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 –  backstoreality Dec 9 '10 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. –  Qiaochu Yuan Dec 9 '10 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 –  backstoreality Dec 9 '10 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. –  Robin Chapman Dec 9 '10 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

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

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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 :) –  backstoreality Dec 9 '10 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. –  backstoreality Dec 9 '10 at 20:29
@Joe: the passage from the finite to the infinite case should be straightforward. –  Qiaochu Yuan Dec 9 '10 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 –  backstoreality Dec 9 '10 at 20:59