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

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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