Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let us define the following sequence of polynomials for every two non-negative integers $i,d$: $$s_i^d(w)=\sum_{j=0}^{d+1} (-1)^j {d+1\choose j} (j+1)^i w^{d+1-j}.$$ Conjecture: The sequence $\{s_d^d(w),s_d^{d-1}(w),\dots,s_d^0(w)\}$ is a Sturm sequence.

An easy corollary of this conjecture is that:

Corollary: The roots of the polynomial $s_i^d(w)$ are all real, simple (and positive).

Any idea how to prove either the Conjecture or (directly) the Corollary? Or, in the worst case, the Corollary for the special case $i=d$?

Note that:

  1. $s_0^d(w)=(w-1)^{d+1}$
  2. $s_i^0(w)=w-2^i$
  3. ${d\over dw} s_i^d(w)=(d+1)!s_i^{d-1}(w)$.
share|cite|improve this question
+1, Now I know Sturm's theorem. – Salech Alhasov Oct 17 '12 at 12:24

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.