Orthogonal Polynomials and Painlevé Equations (Australian Mathematical Society Lecture Series)

Description

The first detailed account of the relationships between Painlevé equations and orthogonal polynomials. It gives clear examples as well as proofs, and there are exercises throughout to help the reader get comfortable with the material. Useful for researchers across both fields and anyone interested in integrable systems and non-linear equations. Contents Preface 1 Introduction 1.1 Orthogonal polynomials on the real line 1.1.1 Pearson equation and semi-classical orthogo- nal polynomials 1.2 Painleve´ equations 1.2.1 The six Painleve´ differential equations 1.2.2 Discrete Painleve´ equations 2 Freud weights and discrete Painleve´ I 2.1 The Freud weight w(x) = e−x4 +tx2 2.2 Asymptotic behavior of the recurrence coefficients 2.3 Unicity of the positive solution of d-PI with x0 = 0 2.4 The Langmuir lattice 2.5 Painleve´ IV 2.6 Orthogonal polynomials on a cross 3 Discrete Painleve´ II 3.1 Orthogonal polynomials on the unit circle 3.1.1 The weight w(θ) = et cos θ 3.1.2 The Ablowitz–Ladik lattice 3.1.3 Painleve´ V and III 3.2 Discrete orthogonal polynomials 3.2.1 Generalized Charlier polynomials 3.2.2 The Toda lattice 3.2.3 Painleve´ V and III 3.3 Unicity of solutions for d-PII 4 Ladder operators 4.1 Orthogonal polynomials with exponential weights 4.2 Riemann–Hilbert problem for orthogonal polynomials 4.3 Proof of the ladder operators 4.4 A modification of the Laguerre polynomials 4.5 Ladder operators for orthogonal polynomials on the linear lattice 4.6 Ladder operators for orthogonal polynomials on a q-lattice 5 Other semi-classical orthogonal polynomials 5.1 Semi-classical extensions of Laguerre polynomials 5.2 Semi-classical extensions of Jacobi polynomials 5.3 Semi-classical extensions of Meixner polynomials 5.4 Semi-classical extensions of Stieltjes–Wigert and q-Laguerre polynomials 5.5 Semi-classical bi-orthogonal polynomials on the unit circle 5.6 Semi-classical extensions of Askey–Wilson polynomials 6 Special solutions of Painleve´ equations 6.1 Rational solutions 6.1.1 Painleve´ II 6.1.2 Painleve´ III 6.1.3 Painleve´ IV 6.1.4 Painleve´ V 6.1.5 Painleve´ VI 6.2 Special function solutions 6.2.1 Painleve´ II 6.2.2 Painleve´ III 6.2.3 Painleve´ IV 6.2.4 Painleve´ V 6.2.5 Painleve´ VI 7 Asymptotic behavior of orthogonal polynomials near critical points 7.1 Painleve´ I 7.2 Painleve´ II 7.3 Painleve´ III 7.4 Painleve´ IV 7.5 Painleve´ V 7.6 Painleve´ VI Appendix Solutions to the exercises References Index