Encyclopedia of special functions : the Askey-Bateman project Volume 1 Univariate orthogonal polynomials / edited by Mourad H. Ismail (University of Central Florida) with assistance by Walter van Assche (KU Leuven, Belgium)

This is the first of three volumes that form the Encyclopedia of Special Functions, an extensive update of the Bateman Manuscript Project. Volume 1 contains most of the material on orthogonal polynomials, from the classical orthogonal polynomials of Hermite, Laguerre and Jacobi to the Askey-Wilson polynomials, which are the most general basic hypergeometric orthogonal polynomials. Separate chapters cover orthogonal polynomials on the unit circle, zeros of orthogonal polynomials and matrix orthogonal polynomials, with detailed results about matrix-valued Jacobi polynomials. A chapter on moment problems provides many examples of indeterminate moment problems. A thorough bibliography rounds off what will be an essential reference 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 The Encyclopedia of Special Functions provides an extensive update of the Bateman Manuscript Project. The three volumes will be indispensable for all scientists who use special functions in their research. Volume 1 provides detailed and up-to-date information on orthogonal polynomials and moment problems.