Encyclopedia of Special Functions: The Askey-Bateman Project: Volume 1 - Univariate Orthogonal Polynomials

Copyright
Contents
List of contributors
Preface
1 Preliminaries
	1.1 Analytic Facts
	1.2 Hypergeometric Functions
	1.3 Summation Theorems and Transformations
	1.4 q-Series
	1.5 Theta Functions
	1.6 Orthogonality
2 General Orthogonal Polynomials
	2.1 Basic Facts
	2.2 Numerators and Quadratures
	2.3 The Spectral Theorem
	2.4 Continued Fractions
	2.5 Modifications of Measures and Recursions
	2.6 Linearization and Connection Relations
	2.7 Addition Theorems
	2.8 Differential Equations
	2.9 Discriminants and Electrostatics
	2.10 Functions of the Second Kind
	2.11 Dual Systems
	2.12 Moment Representations and Determinants
3 Jacobi and Related Polynomials
	3.1 Recursions and Representations
	3.2 Generating Functions
	3.3 Jacobi Functions of the Second Kind
	3.4 Routh–Jacobi Polynomials
	3.5 Ultraspherical (Gegenbauer) Polynomials
	3.6 Chebyshev Polynomials
	3.7 Legendre Polynomials
	3.8 Laguerre and Hermite Polynomials
	3.9 The Complex Hermite Polynomials
	3.10 Hermite Functions
	3.11 Multilinear Generating Functions
	3.12 Integral Representations
	3.13 Asymptotics
	3.14 Relative Extrema of Classical Polynomials
	3.15 The Bessel Polynomials
4 Recursively Defined Polynomials
	4.1 Birth and Death Process Polynomials
	4.2 Polynomials of Pollaczek Type
	4.3 Associated Laguerre and Hermite Polynomials
	4.4 Associated Jacobi Polynomials
	4.5 Sieved Polynomials
5 Wilson and Related Polynomials
	5.1 The Meixner–Pollaczek Polynomials
	5.2 Wilson Polynomials
	5.3 Continuous Dual Hahn Polynomials
	5.4 Continuous Hahn Polynomials
6 Discrete Orthogonal Polynomials
	6.1 Meixner and Charlier Polynomials
	6.2 Hahn, Dual Hahn, and Krawtchouk Polynomials
	6.3 Difference Equations
	6.4 Lommel Polynomials and Related Polynomials
	6.5 An Inverse Operator
	6.6 q-Sturm–Liouville Problems
	6.7 The Al-Salam–Carlitz Polynomials
	6.8 q-Jacobi Polynomials
	6.9 q-Hahn Polynomials
	6.10 A Family of Biorthogonal Rational Functions
7 Some q-Orthogonal Polynomials
	7.1 q-Hermite Polynomials
	7.2 q-Ultraspherical Polynomials
	7.3 Asymptotics
	7.4 Integrals and the Rogers–Ramanujan Identities
	7.5 A Generalization of the Schur Polynomials
	7.6 Associated q-Ultraspherical Polynomials
	7.7 Two Systems of q-Orthogonal Polynomials
8 The Askey–Wilson Family of Polynomials
	8.1 Al-Salam–Chihara Polynomials
	8.2 The Askey–Wilson Polynomials
	8.3 The Askey–Wilson Equation
	8.4 Continuous q-Jacobi Polynomials and Discriminants
	8.5 q-Racah Polynomials
	8.6 Linear and Multilinear Generating Functions
	8.7 Associated Askey–Wilson Polynomials
9 Orthogonal Polynomials on the Unit Circle
	9.1 Definitions and Basic Properties
	9.2 Szego˝ Recurrence Relations and Verblunsky Coefficients
	9.3 Szeg˝ o’s Theory and Its Extensions
	9.4 Zeros of OPUC
	9.5 CMV Matrices – Unitary Analogues of Jacobi Matrices
	9.6 Differential Equations
	9.7 Examples of OPUC
	9.8 Modification of Measures
10 Zeros of Orthogonal Polynomials
	10.1 Introduction
	10.2 General Results on Zeros
	10.3 Jacobi Polynomials
	10.4 Ultraspherical Polynomials
	10.5 Legendre Polynomials
	10.6 Laguerre Polynomials
	10.7 Hermite Polynomials and Functions
	10.8 Other Orthogonal Polynomials
11 The Moment Problem
	C. Berg &J. S. Christiansen 11.1 Hamburger Moment Problems
	11.2 Stieltjes Moment Problems
	11.3 Examples of Indeterminate Moment Problems
12 Matrix-Valued Orthogonal Polynomials and Differential Equations
	12.1 Matrix Polynomials and Matrix Orthogonality
	12.2 Matrix-Valued Orthogonal Polynomials Satisfying Second-Order Differential Equations
13 Some Families of Matrix-Valued Jacobi Orthogonal Polynomials
	13.1 Introduction
	13.2 Spherical Functions
	13.3 Matrix-Valued Spherical Functions Associated to P2(C)
	13.4 The Spherical Functions as Matrix-Valued Hypergeometric Functions
	13.5 Matrix Orthogonal Polynomials Arising from Spherical Functions
	13.6 The Matrix Jacobi Polynomials Arising from Pd(C)
	13.7 Miscellanea
References
Index