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.