Asymptotic and interlacing properties of zeros of exceptional Jacobi and Laguerre polynomials

Abstract

a b s t r a c tIn this paper we state and prove some properties of the zeros of exceptional Jacobi and Laguerre polynomials. Generically, the zeros of exceptional polynomials fall into two classes: the regular zeros, which lie in the interval of orthogonality and the exceptional zeros, which lie outside that interval. We show that the regular zeros have two interlacing properties: one is the natural interlacing between zeros of consecutive polynomials as a consequence of their Sturm-Liouville character, while the other one shows interlacing between the zeros of exceptional and classical polynomials. A Heine-Mehler type formula is provided for the exceptional polynomials, which allows to derive the asymptotic behaviour of their regular zeros for large degree n and fixed codimension m. We also describe the location and the asymptotic behaviour of the m exceptional zeros, which converge for large n to fixed values.

The electrostatic properties of zeros of exceptional Laguerre and Jacobi polynomials and stable interpolation

“…Lemma 2 (Proposition 3.2, Corollary 3.1, Proposition 3.4. in [12]). For α > 0 L I,(α) m,m+n has m simple exceptional zeros in (−∞, 0): z m,n,1 > · · · > z m,n,m , and n simple regular zeros in (0, ∞): x m,n,1 < · · · < x m,n,n .…”

“…Here we have Lemma 4 (Propositions 5.3, 5.4, Corollary 5.1. in [12]). Let us suppose that α, β and m satisfy the condition α + 1 − m − β ̸ ∈ {0, 1, .…”

“…Higher-codimensional families were introduced by S. Odake and R. Sasaki [22]. The location of zeros of exceptional orthogonal polynomials is described by D. Gómez-Ullate, F. Marcellán and R. Milson [12], and the electrostatic interpretation of zeros of X 1 -Jacobi polynomials is given by D. Dimitrov and Yen Chi Lun [3].…”

A class of exactly solvable rationally extended Calogero–Wolfes type 3-body problems

“…After the recent discovery of two new orthogonal polynomials namely the exceptional X 1 Laguerre and X 1 Jacobi (or more general X m Laguerre and X p Jacobi respectively) polynomials [18,19,20,21], a number of one body exactly solvable potentials [22,23,24,25,26,27,28,29,30] as well as Calogero type many particle potentials [31] have been extended rationally whose solutions are obtained in terms of these exceptional orthogonal polynomials (EOPs). Many interesting properties of these extended potentials have been studied [32,33,34,35,36,37,38,39,40,41].…”

Group theoretic approach to rationally extended shape invariant potentials

“…The properties of these X m exceptional orthogonal polynomials have been studied in detail in Ref. [8,9,10,11,12]. After the discovery of these two polynomials the new rationally extended potentials have been obtained whose solutions are in terms of X m EOPs [7].…”