TITLE: Roots of Legendre Polynomial
QUESTION [28 upvotes]: I was wondering if the following properties of the Legendre polynomials are true in general. They hold for the first ten or fifteen polynomials.

Are the roots always simple (i.e., multiplicity $1$)?
Except for low-degree cases, the roots can't be calculated exactly, only approximated (unlike Chebyshev polynomials).
Are roots of the entire family of Legendre Polynomials dense in the interval $[0,1]$ (i.e., it's not possible to find a subinterval, no matter how small, that doesn't contain at least one root of one polynomial)?

If anyone knows of an article/text that proves any of the above, please let me know.  The definition of these polynomials can be found on Wikipedia.

REPLY [10 votes]: The density of the roots of any family of orthogonal polynomials follows from this result:

If $\{p_n\}$ is a family of orthogonal polynomials with roots in $[-1,1]$ and $N(a,b,n)$ represents the number of roots of $p_n$ in $[\cos(b),\cos(a)]$ then
$$\lim_{n\to \infty} \frac{N(a,b,n)}{n} = \frac{b-a}{\pi}$$