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Classical and Quantum Orthogonal Polynomials in One Variable.
The first modern treatment of orthogonal polynomials from the viewpoint of special functions is now available in paperback.
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Cambridge :
Cambridge University Press,
2005.
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| Colección: | Encyclopedia of mathematics and its applications.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Cover; Half Title; Series Page; Title; Copyright; Contents; Foreword; Preface; 1 Preliminaries; 1.1 Hermitian Matrices and Quadratic Forms; 1.2 Some Real and Complex Analysis; 1.3 Some Special Functions; 1.4 Summation Theorems and Transformations; Exercises; 2 Orthogonal Polynomials; 2.1 Construction of Orthogonal Polynomials; 2.2 Recurrence Relations; 2.3 Numerator Polynomials; 2.4 Quadrature Formulas; 2.5 The Spectral Theorem; 2.6 Continued Fractions; 2.7 Modifications of Measures: Christoffel and Uvarov; 2.8 Modifications of Measures: Toda; 2.9 Modification by Adding Finite Discrete Parts.
- 2.10 Modifications of Recursion Coefficients2.11 Dual Systems; Exercises; 3 Differential Equations Discriminants and Electrostatics; 3.1 Preliminaries; 3.2 Differential Equations; 3.3 Applications; 3.4 Discriminants; 3.5 An Electrostatic Equilibrium Problem; 3.6 Functions of the Second Kind; 3.7 Differential Relations and Lie Algebras; Exercises; 4 Jacobi Polynomials; 4.1 Orthogonality; 4.2 Differential and Recursion Formulas; 4.3 Generating Functions; 4.4 Functions of the Second Kind; 4.5 Ultraspherical Polynomials; 4.6 Laguerre and Hermite Polynomials; 4.7 Multilinear Generating Functions.
- 4.8 Asymptotics and Expansions4.9 Relative Extrema of Classical Polynomials; 4.10 The Bessel Polynomials; Exercises; 5 Some Inverse Problems; 5.1 Ultraspherical Polynomials; 5.2 Birth and Death Processes; 5.3 The Hadamard Integral; 5.4 Pollaczek Polynomials; 5.5 A Generalization; 5.6 Associated Laguerre and Hermite Polynomials; 5.7 Associated Jacobi Polynomials; 5.8 The J-Matrix Method; 5.9 The Meixner-Pollaczek Polynomials; Exercises; 6 Discrete Orthogonal Polynomials; 6.1 Meixner Polynomials; 6.2 Hahn, Dual Hahn, and Krawtchouk Polynomials; 6.3 Difference Equations.
- 6.4 Discrete Discriminants6.5 Lommel Polynomials; 6.6 An Inverse Operator; Exercises; 7 Zeros and Inequalities; 7.1 A Theorem of Markov; 7.2 Chain Sequences; 7.3 The Hellmann-Feynman Theorem; 7.4 Extreme Zeros of Orthogonal Polynomials; 7.5 Concluding Remarks; 8 Polynomials Orthogonal on the Unit Circle; 8.1 Elementary Properties; 8.2 Recurrence Relations; 8.3 Differential Equations; 8.4 Functional Equations and Zeros; 8.5 Limit Theorems; 8.6 Modifications of Measures; Exercises; 9 Linearization, Connections and Integral Representations; 9.1 Connection Coefficients.
- 9.2 The Ultraspherical Polynomials and Watson's Theorem9.3 Linearization and Power Series Coefficients; 9.4 Linearization of Products and Enumeration; 9.5 Representations for Jacobi Polynomials; 9.6 Addition and Product Formulas; 9.7 The Askey-Gasper Inequality; Exercises; 10 The Sheffer Classification; 10.1 Preliminaries; 10.2 Delta Operators; 10.3 Algebraic Theory; Exercises; 11 q-Series Preliminaries; 11.1 Introduction; 11.2 Orthogonal Polynomials; 11.3 The Bootstrap Method; 11.4 q-Differences; 12 q-Summation Theorems; 12.1 Basic Definitions; 12.2 Expansion Theorems; 12.3 Bilateral Series.