A First Course on Orthogonal Polynomials: Classical Orthogonal Polynomials and Related Topics

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
Authors
Symbol Description
1. Foundations of the Algebraic Theory
	1.1. The Space P
	1.2. Dual Basis
	1.3. Basic Operations
	1.4. The Leibniz Formula
	1.5. The Formal Stieltjes Series
2. Orthogonal Polynomial Sequences
	2.1. Orthogonal Polynomial Sequences
	2.2. Existence of Orthogonal Polynomial Sequences
	2.3. The Positive Definite Case
	2.4. The Favard Theorem
	2.5. The Christoffel-Darboux Identity
3. Zeros and Gauss-Jacobi Mechanical Quadrature
	3.1. Zeros of Orthogonal Polynomials
	3.2. The Gauss-Jacobi Mechanical Quadrature
	3.3. The Wendroff Theorem
4. The Spectral Theorem
	4.1. The Helly Theorems
	4.2. The Representation Theorem
	4.3. The Spectral Theorem
	4.4. The Unicity of the Spectral Measure
5. The Markov Theorem
	5.1. The Perron-Stieltjes Inversion Formula
	5.2. Associated Polynomials
	5.3. The Markov Theorem
6. Orthogonal Polynomials and Dual Basis
	6.1. Orthogonal Polynomials and Dual Basis
	6.2. The Translation and Homothetic Operators
7. Functional Differential Equation
	7.1. The Functional Differential Equation
	7.2. Regularity Conditions
8. Classical Orthogonal Polynomials: General Properties
	8.1. Definition and Characterizations
	8.2. Classification and Canonical Representatives
9. Functional Equation on Lattices
	9.1. Functional Equation on Lattices
		9.1.1. Preliminary Properties
		9.1.2. The Functional Rodrigues Formula
		9.1.3. The Regularity of u[k]
	9.2. The Regularity Conditions
		9.2.1. The Lattice x(s) = c1q−s + c2qs + c3
		9.2.2. The Lattice x(s) = c4s2 + c5s + c6
	9.3. Two Examples: Racah and Askey-Wilson Polynomials
		9.3.1. The Racah Polynomials
		9.3.2. The Askey-Wilson Polynomials
10. Classical Orthogonal Polynomials: The Positive Definite Case
	10.1. The Positive Definite Case
		10.1.1. Hermite Functional
		10.1.2. Laguerre Functional
		10.1.3. Jacobi Functional
		10.1.4. Bessel Functional
	10.2. Orthogonality of the Bessel Polynomials
	10.3. Explicit Expressions for the Classical Orthogonal Polynomials
		10.3.1. The Rodrigues Formula
		10.3.2. Jacobi Polynomials
		10.3.3. Laguerre Polynomials
		10.3.4. Hermite Polynomials
		10.3.5. Bessel Polynomials
11. Hypergeometric Series
	11.1. The Gamma and Beta Functions
	11.2. Hypergeometric Series
	11.3. The Hypergeometric Function
		11.3.1. Basic Properties
		11.3.2. The Euler Integral Representation
		11.3.3. The Gauss Summation Formula
	11.4. The Pfaff-Saalschütz, Dixon, and Dougall Identities
	11.5. Binomial Sums
A. Locally Convex Spaces
	A.1. Definitions and Basic Properties
	A.2. Fréchet Spaces
	A.3. The Inductive Limit Topology
	A.4. The Weak Dual Topology
Bibliography
Index