TITLE: Recurrence of Legendre polynomial roots/ quadrature points
QUESTION [6 upvotes]: Consider Legendre polynomials $p_n (x)$ on $[-1,1]$. For each $n \in \mathbb{N}$ we denote the zeros of $p_n (x)$ by $\left( x_j ^{(n)} \right) _{j=1} ^n$. 
We know that these roots are distinct, and we know that for all $n\in \mathbb{N}$ then $x_j ^{(n)} \neq x_i ^{(n+1)}$ for all $i$ and $j$. In fact, we even now that they are interlaced i.e. $x_i ^{(n+1)} < x_i ^{(n)}  < x_{i+1} ^{(n+1)}$.
On the other hand, we know for example that for all odd $n=2k+1$ we have that $x_{k+1} ^{(2k+1)} = 0$, and so I arrive at my question:
Question 1: Are there any rules as to if, when, and how roots recur, i.e. $x_i ^{(n)} = x_j ^{(m)}$ for some relevant indices?
Question 2: Apart from the well known Chebyshev polynomials, are there any other orthogonal polynomials with high rate of roots recurrence?

REPLY [6 votes]: It is a conjecture of Stieltjes, apparently still open, see
T.J. Stieltjes, Letter No. 275 of Oct. 2, 1890, in Correspondance d'Hermite et de
Stieltjes, vol 2, Gauthier-Villars, Paris, 1905.
that Legendre polynomials of different degrees have no common roots, except $x=0$ when both degrees are odd.
Laguerre polynomials $L_{n}^{\alpha}$, $\alpha>-1$, orthogonal with respect to the weight function $x^{\alpha}e^{-x}$ on $(0,\infty)$, may have common roots. For instance, as observed in 
K. Driver, K. Jordaan, Stieltjes interlacing of zeros of Laguerre polynomials from different sequences. Indag. Math. (N.S.) 21 (2011), 204-211.
$L_{2}^{23}(x)$ and $L_{4}^{23}(x)$ have $x=30$ as a common root.
It seems also to be an open question how many common
zeros are possible in general for $L_{n}^{\alpha}$ and $L_{n+m}^{\alpha}$, $n\geq 0$ and $m>1$. The corresponding question for Hermite polynomials about a common root other than zero seems to be unanswered as well.
Finally, a classical theorem of Stieltjes states that if $p_{0},p_{1},p_{2},\ldots$ is any sequence of orthogonal polynomials then the zeros of $p_{k}$ and $p_{n}$, $k<n$, are interlacing in the sense that each open interval of the form 
$$(-\infty, z_{1}), (z_{1}, z_{2}), . . . , (z_{k-1}, z_{k}), (z_{k} , \infty)$$
where $z_{1}<z_{2}<\cdots<z_{k}$ are the zeros of $p_{k}$, contains at least one zero of $p_{n}$. It implies that at least $k+1$ zeros of
$p_{n}$ are distinct from the zeros of $p_{k}$, or equivalently $p_{k}$ and $p_{n}$ have at most $\min(k,n-k-1)$ zeros in common. It is proved in
P.C. Gibson, Common zeros of two polynomials in an orthogonal sequence, J. Approx. Theory 105 (2000) 129-132.
that this bound is sharp in the sense that for any $k<n$, there exists a sequence of orthogonal polynomials $q_{0},\ldots,q_{n}$ such that $q_{k}$ and $q_{n}$ have precisely $\min(k,n-k-1)$ zeros in common.