Classical Orthogonal Polynomials of a Discrete Variable

The authors give a concise introduction to the theory of orthogonal polynomials of a discrete variable and present the methods of solving a large class of difference equations of hypergeometric type. They apply the theory to various problems in scientific computing, probability, queuing theory, coding, and information compression. Students and scientists will find a useful textbook in numerical analysis.

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.- 1.4.1 Classification of Polynomials.- 1.4.2 General Properties of Orthogonal 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.- 3.6.1 The q-Analogs of the Hahn, Meixner, Kravchuk, and Charlier Polynomials on the Lattices x(s) = exp(2?s) and x(s) = sinh(2?s).- 3.6.2 The q-Analogs of the Racah and Dual Hahn Polynomials on the Lattices x(s) = cosh(2?s) and x(s) = cos(2?s).- 3.6.3 Tables of Basic Data for q-Analogs.- 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.- 5.1.1 The Three-Dimensional Rotation Group and Its Irreducible Representations.- 5.1.2 Expressing the Generalized Spherical Functions in Terms of the Jacobi and Kravchuk Polynomials.- 5.1.3 Major Properties of Generalized Spherical Functions.- 5.2 Clebsch-Gordan Coefficients and Hahn Polynomials.- 5.2.1 The Tensor Product of the Rotation Group Representations.- 5.2.2 Expressing the Clebsch-Gordan Coefficients in Terms of Hahn Polynomials.- 5.2.3 Main Properties of the Clebsch-Gordan Coefficients..- 5.2.4 Irreducible Tensor Operators. The Wigner-Eckart Theorem.- 5.3 The Wigner 6j-Symbols and the Racah Polynomials.- 5.3.1 The Racah Coefficients and the Wigner 6j-Symbols.- 5.3.2 Expressing the 6j-Symbols Through the Racah Polynomials.- 5.3.3 Main Properties of the 6j-Symbols.- 5.4 The Wigner 9j-Symbols as Orthogonal Polynomials in Two Discrete Variables.- 5.4.1 The 9j-Symbols and the Relation with the Clebsch-Gordan Coefficients.- 5.4.2 The Polynomial Expression for the 9j-Symbols.- 5.4.3 Basic Properties of the Polynomials Related to the 9j-Symbols.- 5.5 The Classical Orthogonal Polynomials of a Discrete Variable in Some Problems of Group Representation Theory.- 5.5.1 The Hahn Polynomials and the Representations of the Rotation Group in the Four-Dimensional Space.- 5.5.2 The Unitary Irreducible Representations of the Lorentz Group SO(l,3) and Hahn Polynomials in an Imaginary Argument.- 5.5.3 The Racah Polynomials and the Representations of the Group SU(3).- 5.5.4 The Charlier Polynomials and Representations of the Heisenberg-Weyl Group.- 6. Hyperspherical Harmonics.- 6.1 Spherical Coordinates in a Euclidean Space.- 6.1.1 Setting up Spherical Coordinates.- 6.1.2 A Metric and Elementary Volume.- 6.1.3 The Laplace Operator.- 6.1.4 A Graphical Approach.- 6.2 Solution of the n-Dimensional Laplace Equation in Spherical Coordinates.- 6.2.1 Separation of Variables.- 6.2.2 Hyperspherical Harmonics.- 6.2.3 Illustrative Examples.- 6.3 Transformation of Harmonics Derived in Different Spherical Coordinates.- 6.3.1 Transpositions and Transplants.- 6.3.2 The T-Coefficients for a Transplant Involving Closed Nodes.- 6.3.3 Open Nodes.- 6.4 Solution of the Schrödinger Equation for the n-Dimensional Harmonic Oscillator.- 6.4.1 Wave Functions of the Harmonic Oscillator in n Dimensions.- 6.4.2 Transformation Between Wave Functions of the Oscillator in Cartesian and Spherical Coordinates.- 6.4.3 The T-Coefficients as the 3nj-Symbols of SU(1,1)..- 6.4.4 Matrix Elements of SU(1,1).- 6.4.5 Harmonic Oscillator and Matrix Elements of the Heisenberg-Weyl Group N(3).- Addendum to Chapter 6.