# Recurrence of Legendre polynomial roots/ quadrature points

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?

• Does not recurrence relation among Legendre polynomial's (or Chebyshev, Hermite etc) and ODE they satisfy give you an information about a relation between consecutive zeros? Sep 5, 2016 at 2:42
• I assume I don't know ALL the relevant recurrence relations, but from the ones I do I couldn't deduce to recurrence of zeros. I'd be happy if you will prove me wrong. Sep 5, 2016 at 5:14
• Using [iteration][1] we can using additionaly Newton's method approximate the non zero real roots of the Legendre polynomial.A bound can be found for the smallest root in the link. [1]: math.stackexchange.com/questions/4738155/… Nov 25, 2023 at 15:13

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.

Numerical evidence suggests that the nonzero roots of the Legendre polynomials do not repeat. The numerical experiment I performed is simple: I gathered all of the nonzero roots of the first 100 Legendre polynomials. I sorted them in ascending order. I then graphed the first order difference of this sorted vector with logarithmic scaling in the vertical direction. Here is the graphical output.

Its a bit hard to see, but there are no holes in this figure, which suggests that all 5000 nonzero roots of the first 100 Legendre polynomials are distinct.

all_roots=[];