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…mehr
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.
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.
Inhaltsangabe
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.
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.
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