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

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