Exceptional Orthogonal Polynomials and Rational Solutions to Painlevé Equations

Abstract

These are the lecture notes for a course on exceptional polynomials taught at the AIMS-Volkswagen Stiftung Workshop on Introduction to Orthogonal Polynomials and Applications that took place in Douala (Cameroon) from October 5-12, 2018. They summarize the basic results and construction of exceptional poynomials, developed over the past ten years. In addition, some new results are presented on the construction of rational solutions to Painlevé equation PIV and its higher order generalizations that belong to the A2n -Painlevé hierarchy. The construction is based on dressing chains of Schrödinger operators with potentials that are rational extensions of the harmonic oscillator. Some of the material presented here (Sturm-Liouville operators, classical orthogonal polynomials, Darboux-Crum transformations, etc.) are classical and can be found in many textbooks, while some results (genus, interlacing and cyclic Maya diagrams) are new and presented for the first time in this set of lecture notes. (2000). Primary 33C45; Secondary 34M55. Mathematics Subject Classification

Exceptional Hahn and Jacobi polynomials with an arbitrary number of continuous parameters

“…In Lemma 3, we prove that the sequence of orthogonal polynomials with respect to the measure 𝜈 ,𝑈 𝐹 𝑎,𝑏, N (18) and ℎ 𝖺,𝖻,𝑁;,𝐹 𝑛 are dual sequences.…”

“…There are several applications of exceptional polynomials in mathematical physics (they allow the explicit computation of bound states of rational extensions of classical quantum-mechanical potentials, and also appear in connection with superintegrable systems 15 or shape-invariant potentials 16 ), and in the construction of rational solutions for certain instances of the Painlevé equation. 17,18 In all the examples appeared before 2015 apart from the parameters associated to the classical and classical discrete weights, only discrete parameters appear in the construction of each exceptional family. This scenario changed in 2015, when Bagchi et al 19 and then Grandati and Quesne 20 constructed exceptional Jacobi polynomials depending on one continuous parameter.…”

“…Under mild conditions on the parameters, we prove in Ref. 22 that the polynomials 𝑞 𝑎,𝑏, N;,𝑈 𝐹 𝑛 (𝑥), 𝑛 ≥ 0, are orthogonal with respect to the measure 𝜈 ,𝑈 𝐹 𝑎,𝑏, N (see (18)). In particular, that is the case when 𝑁 is a positive integer and the measure 𝜈 ,𝑈 𝐹 𝑎,𝑏, N is positive; the polynomials (94) have then positive norm when 𝑛 = 0, … , 𝑁 − 𝑛 𝐹 .…”

A family of fourth-order superintegrable systems with rational potentials related to Painlevé VI

“…The proceeding discussion describes the construction of the integrals of motion and it remains only to show by direct computation that the leading order-terms of L 1 (9). Theorem 1 (Gravel [11]).…”

“…These rational extension of known exactly-solvable or superintegrable systems have been investigated by several authors and have been linked to potential associated with Painlevé transcendents [6,9,18,26,28,29]. In a previous paper by the authors [24], a system associated with exceptional Jacobi polynomials was constructed, shown to be superintegrable and identified in the classification by Escobar-Ruiz, López Vieyra and Winternitz [3] to be associated with Painlevé VI.…”