Classical Orthogonal Polynomials of a Discrete Variable / by Arnold F. Nikiforov, Vasilii B. Uvarov, Sergei K. Suslov.
While classical orthogonal polynomials appear as solutions to hypergeometric differential equations, those of a discrete variable emerge as solutions of difference equations of hypergeometric type on lattices. The authors present a concise introduction to this theory, presenting at the same time met...
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| Format: | eBook |
| Language: | English |
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Berlin, Heidelberg :
Springer Berlin Heidelberg,
1991.
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| Series: | Springer series in computational physics.
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Table of Contents:
- 1. Classical Orthogonal Polynomials
- 1.1 An Equation of Hypergeometric Type
- 1.2 Polynomials of Hypergeometric Type and Their Derivatives. The Rodrigues Formula
- 1.3 The Orthogonality Property
- 1.4 The Jacobi, Laguerre, and Hermite Polynomials
- 1.5 Classical Orthogonal Polynomials as Eigenfunctions of Some Eigenvalue Problems
- 2. Classical Orthogonal Polynomials of a Discrete Variable
- 2.1 The Difference Equation of Hypergeometric Type
- 2.2 Finite Difference Analogs of Polynomials of Hypergeometric Type and of Their Derivatives. The Rodrigues Type Formula
- 2.3 The Orthogonality Property
- 2.4 The Hahn, Chebyshev, Meixner, Kravchuk, and Charlier Polynomials
- 2.5 Calculation of Main Characteristics
- 2.6 Asymptotic Properties. Connection with the Jacobi, Laguerre, and Hermite Polynomials
- 2.7 Representation in Terms of Generalized Hypergeometric Functions
- 3. Classical Orthogonal Polynomials of a Discrete Variable on Nonuniform Lattices
- 3.1 The Difference Equation of Hypergeometric Type on a Nonuniform Lattice
- 3.2 The Difference Analogs of Hypergeometric Type Polynomials. The Rodrigues Formula
- 3.3 The Orthogonality Property
- 3.4 Classification of Lattices
- 3.5 Classification of Polynomial Systems on Linear and Quadratic Lattices. The Racah and the Dual Hahn Polynomials
- 3.6 q-Analogs of Polynomials Orthogonal on Linear and Quadratic Lattices
- 3.7 Calculation of the Leading Coefficients and Squared Norms. Tables of Data
- 3.8 Asymptotic Properties of the Racah and Dual Hahn Polynomials
- 3.9 Construction of Some Orthogonal Polynomials on Nonuniform Lattices by Means of the Darboux-Christoffel Formula
- 3.10 Continuous Orthogonality
- 3.11 Representation in Terms of Hypergeometric and q-Hypergeometric Functions
- 3.12 Particular Solutions of the Hypergeometric Type Difference Equation
- Addendum to Chapter 3
- 4. Classical Orthogonal Polynomials of a Discrete Variable in Applied Mathematics
- 4.1 Quadrature Formulas of Gaussian Type
- 4.2 Compression of Information by Means of the Hahn Polynomials
- 4.3 Spherical Harmonics Orthogonal on a Discrete Set of Points
- 4.4 Some Finite-Difference Methods of Solution of Partial Differential Equations
- 4.5 Systems of Differential Equations with Constant Coefficients. The Genetic Model of Moran and Some Problems of the Queueing Theory
- 4.6 Elementary Applications to Probability Theory
- 4.7 Estimation of the Packaging Capacity of Metric Spaces
- 5. Classical Orthogonal Polynomials of a Discrete Variable and the Representations of the Rotation Group
- 5.1 Generalized Spherical Functions and Their Relations with Jacobi and Kravchuk Polynomials
- 5.2 Clebsch-Gordan Coefficients and Hahn Polynomials
- 5.3 The Wigner 6j-Symbols and the Racah Polynomials
- 5.4 The Wigner 9j-Symbols as Orthogonal Polynomials in Two Discrete Variables
- 5.5 The Classical Orthogonal Polynomials of a Discrete Variable in Some Problems of Group Representation Theory
- 6. Hyperspherical Harmonics
- 6.1 Spherical Coordinates in a Euclidean Space
- 6.2 Solution of the n-Dimensional Laplace Equation in Spherical Coordinates
- 6.3 Transformation of Harmonics Derived in Different Spherical Coordinates
- 6.4 Solution of the Schrödinger Equation for the n-Dimensional Harmonic Oscillator
- Addendum to Chapter 6.