Classical Orthogonal Polynomials of a Discrete Variable

ISBN-13: 9783642747502 / Angielski / Miękka / 2012 / 374 str.

Arnold F. Nikiforov; Sergei K. Suslov; Vasilii B. Uvarov

Opis

Mathematical modelling of many physical processes involves rather complex dif ferential, integral, and integro-differential equations which can be solved directly only in a number of cases. Therefore, as a first step, an original problem has to be considerably simplified in order to get a preliminary knowledge of the most important qualitative features of the process under investigation and to estimate the effect of various factors. Sometimes a solution of the simplified problem can be obtained in the analytical form convenient for further investigation. At this stage of the mathematical modelling it is useful to apply various special functions. Many model problems of atomic, molecular, and nuclear physics, electrody namics, and acoustics may be reduced to equations of hypergeometric type, a(x)y" + r(x)y' + AY = 0, (0.1) where a(x) and r(x) are polynomials of at most the second and first degree re spectively and A is a constant E7, AI, N18]. Some solutions of (0.1) are functions extensively used in mathematical physics such as classical orthogonal polyno mials (the Jacobi, Laguerre, and Hermite polynomials) and hypergeometric and confluent hypergeometric functions.

Spis treści

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.

O autorze

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.