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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. ${d\over dw} s_i^d(w)=(d+1)!s_i^{d-1}(w)$.
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+1, Now I know Sturm's theorem. –  Salech Alhasov Oct 17 '12 at 12:24

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