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.
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A leading authority on orthogonal polynomials details their relationships with Painlevé equations with clear proofs, examples, and exercises.
| Preface |
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xi | |
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1 | (12) |
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1.1 Orthogonal polynomials on the real line |
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1 | (7) |
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1.1.1 Pearson equation and semi-classical orthogonal polynomials |
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4 | (4) |
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8 | (5) |
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1.2.1 The six Painleve differential equations |
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8 | (1) |
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1.2.2 Discrete Painleve equations |
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9 | (4) |
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2 Freud weights and discrete Painleve I |
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13 | (14) |
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2.1 The Freud weight w(x) = e-x+tx2 |
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13 | (3) |
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2.2 Asymptotic behavior of the recurrence coefficients |
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16 | (1) |
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2.3 Unicity of the positive solution of d-PI with xo = 0 |
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17 | (4) |
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21 | (2) |
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23 | (1) |
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2.6 Orthogonal polynomials on a cross |
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24 | (3) |
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27 | (23) |
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3.1 Orthogonal polynomials on the unit circle |
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27 | (7) |
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3.1.1 The weight w(θ) = et cosθ |
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28 | (3) |
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3.1.2 The Ablowitz--Ladik lattice |
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31 | (1) |
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32 | (2) |
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3.2 Discrete orthogonal polynomials |
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34 | (12) |
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3.2.1 Generalized Charlier polynomials |
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36 | (7) |
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43 | (1) |
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44 | (2) |
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3.3 Unicity of solutions for d-PII |
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46 | (4) |
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50 | (14) |
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4.1 Orthogonal polynomials with exponential weights |
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50 | (3) |
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4.2 Riemann-Hilbert problem for orthogonal polynomials |
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53 | (2) |
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4.3 Proof of the ladder operators |
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55 | (2) |
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4.4 A modification of the Laguerre polynomials |
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57 | (3) |
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4.5 Ladder operators for orthogonal polynomials on the linear lattice |
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60 | (1) |
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4.6 Ladder operators for orthogonal polynomials on a q-lattice |
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61 | (3) |
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5 Other semi-classical orthogonal polynomials |
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64 | (19) |
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5.1 Semi-classical extensions of Laguerre polynomials |
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64 | (1) |
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5.2 Semi-classical extensions of Jacobi polynomials |
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65 | (1) |
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5.3 Semi-classical extensions of Meixner polynomials |
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66 | (4) |
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5.4 Semi-classical extensions of Stieltjes--Wigert and q-Laguerre polynomials |
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70 | (4) |
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5.5 Semi-classical bi-orthogonal polynomials on the unit circle |
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74 | (5) |
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5.6 Semi-classical extensions of Askey--Wilson polynomials |
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79 | (4) |
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6 Special solutions of Painleve equations |
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83 | (32) |
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83 | (20) |
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83 | (5) |
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88 | (3) |
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91 | (7) |
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98 | (4) |
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102 | (1) |
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6.2 Special function solutions |
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103 | (12) |
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103 | (3) |
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106 | (2) |
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108 | (1) |
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109 | (3) |
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112 | (3) |
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7 Asymptotic behavior of orthogonal polynomials near critical points |
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115 | (32) |
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119 | (10) |
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129 | (7) |
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136 | (1) |
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137 | (4) |
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141 | (5) |
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146 | (1) |
| Appendix Solutions to the exercises |
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147 | (20) |
| References |
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167 | (10) |
| Index |
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177 | |
Walter Van Assche is a professor of mathematics at the Katholieke Universiteit Leuven, Belgium, and presently the Chair of the SIAM Activity Group on Orthogonal Polynomials and Special Functions (OPSF). He is an expert in orthogonal polynomials, special functions, asymptotics, approximation, and recurrence relations.
Lisainfo e-raamatute kohta