ZERO DISTRIBUTION OF SEQUENCES OF CLASSICAL ORTHOGONAL POLYNOMIALS

In this paper, we study the zero distribution of sequences of Jacobi, Laguerre, and Hermite polynomials. Our approach is based on identifying these orthogonal polynomials with certain Fekete polynomials defined below, and using monotonicity properties of the zeros of the polynomials. Let E ⊂ R be a closed set that consists of finitely many intervals. Let w : E→ [0,∞) be a weight function such that w(x) > 0, x ∈ Int(E), and |x|w(x)→ 0 as |x| →∞, x ∈ E, if E is unbounded. Consider the function


Introduction
In this paper, we study the zero distribution of sequences of Jacobi, Laguerre, and Hermite polynomials.Our approach is based on identifying these orthogonal polynomials with certain Fekete polynomials defined below, and using monotonicity properties of the zeros of the polynomials.
Let E ⊂ R be a closed set that consists of finitely many intervals.Let w : E → [0,∞) be a weight function such that w(x) > 0, x ∈ Int(E), and |x|w(x) → 0 as |x| → ∞, x ∈ E, if E is unbounded.Consider the function V n x 1 ,...,x n := 1≤i< j≤n w x i w x j x j − x i , (1.1) It can be shown that V n attains its maximum for some set Ᏺ n = {x i } n i=1 ⊂ E called nth weighted Fekete set or simply Fekete set.We introduce the following notation: if µ is a measure, its logarithmic potential U µ (z) is defined by logarithmic energy I w (µ) := log 1 w(z)w(t)|z − t| dµ(z)dµ(t) (1.3) over all probability Borel measures supported on E. The support of the measure µ w will be denoted by S w .The asymptotic distribution of Fekete points is known (see [4, Chapter III, Theorem 1.3]).
Theorem 1.1.Let ν Ᏺn be the discrete measure that has mass 1/n at each Fekete point that is, lim n→∞ ν Ᏺn = µ w in the weak-star topology of measures.Furthermore, if F n is the nth degree monic polynomial with zero set Ᏺ n , uniformly on compact subsets of C \ S w .
We will assume that w(x) = 0 when x ∈ E \ Int(E) and x is finite.This condition implies that every Fekete set Ᏺ n ⊂ Int(E).Consequently, the partial derivatives of log(V 2 n ) vanish at the Fekete points: (1.6) In Section 2, we study the zero distribution of Jacobi polynomials P (αn,βn) n with parameters α n > 0 and β n > 0 that satisfy lim n→∞ α n /n=2α>0 and lim n→∞ β n /n = 2β > 0.
In Section 4, we obtain the zero distribution of the Hermite polynomials H n .Asymptotics and zero distribution of classical orthogonal polynomials have been studied in [1,2,3,5].Here, we extend these results using a simple method that works for all classical orthogonal polynomials.

Zero distribution of Jacobi polynomials
The Jacobi weight w α,β (x) is defined by Plamen Simeonov 987 with positive α and β.The corresponding extremal measure is given by [4, Chapter IV, Section 5] with support [4, Chapter IV, Section 1] where and q n,α,β denote the orthonormal polynomial of degree n and the monic orthogonal polynomial of degree n, respectively, with respect to the weight w α,β .Let denote the discrete probability measure with mass 1/n at each zero of P (α,β) n .Here, δ(x) denotes the discrete probability measure with support x (the point mass at x).
We first show that the Fekete polynomials for Jacobi weights w α,β with α > 0 and β > 0 are, in fact, Jacobi polynomials.

.11)
If α = ∞ and β is finite, the limit of the measures ν n,αn,βn is the point mass at −1.
If α is finite and β = ∞, the limit of the measures ν n,αn,βn is the point mass at 1.

Zero distribution of the Hermite polynomials
The monic Hermite polynomials H n are orthogonal with respect to the weight w(x) = e −x 2 , x ∈ R. Furthermore, y = H n satisfies the differential equation The corresponding extremal measure µ w is given by (see [4, Chapter IV, Theorem 5.1]), To determine the relationship between the zeros of the Hermite polynomials and the Fekete sets for the weight w(x) = e −x 2 , we set w(x) = e −x 2 in (1.1).Since w (x)/w(x) = −2x, (1.6) yields 4(n − 1)x i F n x i − F n x i = 0, i= 1,...,n.(4.7)

Theorem 4 . 1 .
For every n ≥ 1, let ν n denote the discrete probability measure having mass 1/n at each zero z i,n of the Hermite polynomial H n .Then, µ w , n−→ ∞.