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My research has brought me to the following, very general problem.

Given a fixed, but arbitrary, natural number, $\displaystyle v$, consider the following family of polynomials: The $\displaystyle (n-1)^{th}$ derivative of

$$\displaystyle (1-x^2)^{v+n} \ \ \forall n \in \mathbb{N} $$

I would like to prove (or disprove) that the roots of this entire family of polynomials forms a dense subset of the interval $\displaystyle [0,1]$ for any value of $\displaystyle v$ (I am not interested in roots outside the interval $\displaystyle [0,1]$).

In other words, given any subinterval, $\displaystyle [a,b]$,no mater how small, at least one of these polynomials has at least one root in the interval $\displaystyle [a,b]$ (for any fixed value of $\displaystyle v$).

I realize my question is very general and will happily accept any partial solutions.

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    Presumably this is related to your previous question about roots of Legendre polynomials; one technique that might be adaptable to this particular variant of the question is to prove minimal-separation results for the roots. I don't know if I've seen this even for Legendre polynomials, but it ought to be well within reach; lower bounds on the second derivative between two adjacent roots would force a separation between those roots, and those bounds might be accessible by exploiting the differential equation and the Sturm-iness...2010-12-02
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    There's a proof of the interlacing of Legendre polynomial roots in Chihara; have you by any chance been able to see the book?2010-12-03

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