There are a number of intriguing connections between Painlevé equations and orthogonal polynomials, and this book is one of the first to provide an introduction to these. Researchers in integrable systems and non-linear equations will find the many explicit examples where Painlevé equations appear in mathematical analysis very useful. Those interested in the asymptotic behavior of orthogonal polynomials will also find the description of Painlevé transcendants and their use for local analysis near certain critical points helpful to their work. Rational solutions and special function solutions of Painlevé equations are worked out in detail, with a survey of recent results and an outline of their close relationship with orthogonal polynomials. Exercises throughout the book help the reader to get to grips with the material. The author is a leading authority on orthogonal polynomials, giving this work a unique perspective on Painlevé equations.
Author(s): Walter Van Assche
Series: Australian Mathematical Society Lecture Series 27
Publisher: Cambridge University Press
Year: 2017
Language: English
Pages: 193
Contents......Page 8
Preface......Page 11
1.1 Orthogonal polynomials on the real line......Page 14
1.1.1 Pearson equation and semi-classical orthogo- nal polynomials......Page 17
1.2.1 The six Painleve´ differential equations......Page 21
1.2.2 Discrete Painleve´ equations......Page 22
2.1 The Freud weight w(x) = e−x4 +tx2......Page 26
2.2 Asymptotic behavior of the recurrence coefficients......Page 29
2.3 Unicity of the positive solution of d-PI with x0 = 0......Page 30
2.4 The Langmuir lattice......Page 34
2.5 Painleve´ IV......Page 36
2.6 Orthogonal polynomials on a cross......Page 37
3.1 Orthogonal polynomials on the unit circle......Page 40
3.1.1 The weight w(θ) = et cos θ......Page 41
3.1.2 The Ablowitz–Ladik lattice......Page 44
3.1.3 Painleve´ V and III......Page 45
3.2 Discrete orthogonal polynomials......Page 47
3.2.1 Generalized Charlier polynomials......Page 49
3.2.2 The Toda lattice......Page 56
3.2.3 Painleve´ V and III......Page 57
3.3 Unicity of solutions for d-PII......Page 59
4.1 Orthogonal polynomials with exponential weights......Page 63
4.2 Riemann–Hilbert problem for orthogonal polynomials......Page 66
4.3 Proof of the ladder operators......Page 68
4.4 A modification of the Laguerre polynomials......Page 70
4.5 Ladder operators for orthogonal polynomials on the linear lattice......Page 73
4.6 Ladder operators for orthogonal polynomials on a q-lattice......Page 74
5.1 Semi-classical extensions of Laguerre polynomials......Page 77
5.2 Semi-classical extensions of Jacobi polynomials......Page 78
5.3 Semi-classical extensions of Meixner polynomials......Page 79
5.4 Semi-classical extensions of Stieltjes–Wigert and q-Laguerre polynomials......Page 83
5.5 Semi-classical bi-orthogonal polynomials on the unit circle......Page 87
5.6 Semi-classical extensions of Askey–Wilson polynomials......Page 92
6.1.1 Painleve´ II......Page 96
6.1.2 Painleve´ III......Page 101
6.1.3 Painleve´ IV......Page 104
6.1.4 Painleve´ V......Page 111
6.1.5 Painleve´ VI......Page 115
6.2.1 Painleve´ II......Page 116
6.2.2 Painleve´ III......Page 119
6.2.3 Painleve´ IV......Page 121
6.2.4 Painleve´ V......Page 122
6.2.5 Painleve´ VI......Page 125
7 Asymptotic behavior of orthogonal polynomials near critical points......Page 128
7.1 Painleve´ I......Page 132
7.2 Painleve´ II......Page 142
7.3 Painleve´ III......Page 149
7.4 Painleve´ IV......Page 150
7.5 Painleve´ V......Page 154
7.6 Painleve´ VI......Page 159
Appendix Solutions to the exercises......Page 160
References......Page 180
Index......Page 190
