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Krall-type orthogonal polynomials and integrable isomonodromic deformations

Luc Haine

Abstract

Krall-type polynomials are orthogonal polynomials for a Stieltjes' measure obtained by adding jumps at the boundary of the interval of orthogonality of either the generalized Laguerre polynomials or the Jacobi polynomials. We show that both the recurrence relations and the second order linear differential equations defining these polynomials, are explicitly determined in terms of specific solutions of some integrable systems. When there is only one jump, we are led to integrable cases of the Painlevé III or the Painlevé V equation. In the case of two jumps, first studied by Koornwinder, we obtain a new integrable system of partial differential equations of Schlesinger type. When the jumps are equal and the starting polynomials are the Gegenbauer polynomials, this system reduces to an integrable case of the Painlevé V equation.

Krall-type orthogonal polynomials and integrable isomonodromic deformations

Abstract

Krall-type polynomials are orthogonal polynomials for a Stieltjes' measure obtained by adding jumps at the boundary of the interval of orthogonality of either the generalized Laguerre polynomials or the Jacobi polynomials. We show that both the recurrence relations and the second order linear differential equations defining these polynomials, are explicitly determined in terms of specific solutions of some integrable systems. When there is only one jump, we are led to integrable cases of the Painlevé III or the Painlevé V equation. In the case of two jumps, first studied by Koornwinder, we obtain a new integrable system of partial differential equations of Schlesinger type. When the jumps are equal and the starting polynomials are the Gegenbauer polynomials, this system reduces to an integrable case of the Painlevé V equation.
Paper Structure (7 sections, 24 theorems, 280 equations)

This paper contains 7 sections, 24 theorems, 280 equations.

Table of Contents

  1. Introduction
  2. Semi-classical orthogonal OP and Toda master symmetries
  3. Krall-Laguerre type polynomials and $P_{III}$
  4. Krall-Jacobi type polynomials and $P_V$
  5. Koornwinder's polynomials and a new integrable Schlesinger system of partial differential equations
  6. Koornwinder's formulas for Krall-type polynomials
  7. Appendix

Key Result

Theorem 1.1

The recurrence relation re and the Laguerre differential equation lae for (a) the Krall-Laguerre type polynomials, with weight distribution (b) the Krall-Jacobi type polynomials, with weight distribution (c) the Krall-Gegenbauer type polynomials with weight distribution are completely determined by a sequence of solutions $q_n(t), n\geq 1$, of integrable cases of the $P_{III}$ or the $P_V$ equa

Theorems & Definitions (53)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 3.1
  • Remark 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 43 more