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Rational Solutions of the Fifth Painlevé Equation. Generalised Laguerre Polynomials

Peter A. Clarkson, Clare Dunning

Abstract

In this paper rational solutions of the fifth Painlevé equation are discussed. There are two classes of rational solutions of the fifth Painlevé equation, one expressed in terms of the generalised Laguerre polynomials, which are the main subject of this paper, and the other in terms of the generalised Umemura polynomials. Both the generalised Laguerre polynomials and the generalised Umemura polynomials can be expressed as Wronskians of Laguerre polynomials specified in terms of specific families of partitions. The properties of the generalised Laguerre polynomials are determined and various differential-difference and discrete equations found. The rational solutions of the fifth Painlevé equation, the associated $σ$-equation and the symmetric fifth Painlevé system are expressed in terms of generalised Laguerre polynomials. Non-uniqueness of the solutions in special cases is established and some applications are considered. In the second part of the paper, the structure of the roots of the polynomials are investigated for all values of the parameter. Interesting transitions between root structures through coalescences at the origin are discovered, with the allowed behaviours controlled by hook data associated with the partition. The discriminants of the generalised Laguerre polynomials are found and also shown to be expressible in terms of partition data. Explicit expressions for the coefficients of a general Wronskian Laguerre polynomial defined in terms of a single partition are given.

Rational Solutions of the Fifth Painlevé Equation. Generalised Laguerre Polynomials

Abstract

In this paper rational solutions of the fifth Painlevé equation are discussed. There are two classes of rational solutions of the fifth Painlevé equation, one expressed in terms of the generalised Laguerre polynomials, which are the main subject of this paper, and the other in terms of the generalised Umemura polynomials. Both the generalised Laguerre polynomials and the generalised Umemura polynomials can be expressed as Wronskians of Laguerre polynomials specified in terms of specific families of partitions. The properties of the generalised Laguerre polynomials are determined and various differential-difference and discrete equations found. The rational solutions of the fifth Painlevé equation, the associated -equation and the symmetric fifth Painlevé system are expressed in terms of generalised Laguerre polynomials. Non-uniqueness of the solutions in special cases is established and some applications are considered. In the second part of the paper, the structure of the roots of the polynomials are investigated for all values of the parameter. Interesting transitions between root structures through coalescences at the origin are discovered, with the allowed behaviours controlled by hook data associated with the partition. The discriminants of the generalised Laguerre polynomials are found and also shown to be expressible in terms of partition data. Explicit expressions for the coefficients of a general Wronskian Laguerre polynomial defined in terms of a single partition are given.
Paper Structure (21 sections, 20 theorems, 248 equations, 11 figures, 5 tables)

This paper contains 21 sections, 20 theorems, 248 equations, 11 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Partitions
  3. Generalised Laguerre polynomials
  4. Rational solutions of P$_{\rm V}$
  5. Classification of rational solutions of P$_{\rm V}$
  6. Non-uniqueness of rational solutions of P$_{\rm V}$
  7. Rational solutions of the P$_{\rm V}$ $\sigma$-equation
  8. Hamiltonian structure
  9. Classification of rational solutions of S$_{\rm V}$
  10. Non-uniqueness of rational solutions of S$_{\rm V}$
  11. Applications
  12. Probability density functions associated with the Laguerre unitary ensemble
  13. Joint moments of the characteristic polynomial of CUE random matrices
  14. Rational solutions of the symmetric P$_{\rm V}$ system
  15. Non-uniqueness of rational solutions of sP$_{\rm V}$
  16. ...and 6 more sections

Key Result

Lemma 3.2

The generalised Laguerre polynomial $T_{m,n}^{(\mu)}(z)$ can also be written as the Wronskian

Figures (11)

  • Figure 2.1: The Young diagrams including hook length corresponding to (a) $\bold{\lambda}=(4^2,2,1^3)$ and its core (b) $\overline{\bold{\lambda}}=(2,1)$, and corresponding abacus diagrams (c) and (d).
  • Figure 7.1: The abaci of $\bold{\lambda}_{k,m,n}$.
  • Figure 8.1: The roots of $T_{6,4}^{(\mu)}(\tfrac{1}{2}z^2)$ for various $\mu$.
  • Figure 8.2: Blocks formed by the zeros of $T_{m,n}^{(\mu)}(\tfrac{1}{2}z^2)$.
  • Figure 8.3: The roots of $T_{5,3}^{(\mu)}(\tfrac{1}{2}z^2)$ for $\mu \in [-7, -{\frac{16}{5}} ].$
  • ...and 6 more figures

Theorems & Definitions (72)

  • Definition 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3
  • Definition 3.4
  • Lemma 3.5
  • proof
  • Definition 3.6
  • Remark 3.7
  • Lemma 3.8
  • ...and 62 more