Lectures on Orthogonal Polynomials and Special Functions (London Mathematical Society Lecture Note Series, Series Number 464)

Written by experts in their respective fields, this collection of pedagogic surveys provides detailed insight and background into five separate areas at the forefront of modern research in orthogonal polynomials and special functions at a level suited to graduate students. A broad range of topics are introduced including exceptional orthogonal polynomials, q-series, applications of spectral theory to special functions, elliptic hypergeometric functions, and combinatorics of orthogonal polynomials. Exercises, examples and some open problems are provided. The volume is derived from lectures presented at the OPSF-S6 Summer School at the University of Maryland, and has been carefully edited to provide a coherent and consistent entry point for graduate students and newcomers. Copyright Contents Contributors Preface 1 Exceptional Orthogonal Polynomials via Krall Discrete Polynomials 1.1 Background on classical and classical discrete polynomials 1.1.1 Weights on the real line 1.1.2 The three-term recurrence relation 1.1.3 The classical orthogonal polynomial families 1.1.4 Second-order differential operator 1.1.5 Characterizations of the classical families of orthogonal polynomials 1.1.6 The classical families and the basic quantum models 1.1.7 The classical discrete families 1.2 The Askey tableau. Krall and exceptional polynomials. Darboux Transforms 1.2.1 The Askey tableau 1.2.2 Krall and exceptional polynomials 1.2.3 Krall polynomials 1.2.4 Darboux transforms 1.3 D-operators 1.3.1 D-operators 1.3.2 D-operators on the stage 1.3.3 D-operators of type 2 1.4 Constructing Krall polynomials by using D-operators 1.4.1 Back to the orthogonality 1.4.2 Krall–Laguerre polynomials 1.4.3 Krall discrete polynomials 1.5 First expansion of the Askey tableau. Exceptional polynomials: discrete case 1.5.1 Comparing the Krall continuous and discrete cases (roughly speaking): Darboux transform 1.5.2 First expansion of the Askey tableau 1.5.3 Exceptional polynomials 1.5.4 Constructing exceptional discrete polynomials by using duality 1.6 Exceptional polynomials: continuous case. Second expansion of the Askey tableau 1.6.1 Exceptional Charlier polynomials: admissibility 1.6.2 Exceptional Hermite polynomials by passing to the limit 1.6.3 Exceptional Meixner and Laguerre polynomials 1.6.4 Second expansion of the Askey tableau 1.7 Appendix: Symmetries for Wronskian type determinants whose entries are classical and classical discrete orthogonal polynomials References 2 A Brief Review of q-Series 2.1 Introduction 2.2 Notation and q-operators 2.3 q-Taylor series 2.4 Summation theorems 2.5 Transformations 2.6 q-Hermite polynomials 2.7 The Askey–Wilson polynomials 2.8 Ladder operators and Rodrigues formulas 2.9 Identities and summation theorems 2.10 Expansions 2.11 Askey–Wilson expansions 2.12 A q-exponential function References 3 Applications of Spectral Theory to Special Functions 3.1 Introduction 3.2 Three-term recurrences in l2(Z) 3.2.1 Exercises 3.3 Three-term recurrence relations and orthogonal polynomials 3.3.1 Orthogonal polynomials 3.3.2 Jacobi operators 3.3.3 Moment problems 3.3.4 Exercises 3.4 Matrix-valued orthogonal polynomials 3.4.1 Matrix-valued measures and related polynomials 3.4.2 The corresponding Jacobi operator 3.4.3 The resolvent operator 3.4.4 The spectral measure 3.4.5 Exercises 3.5 More on matrix weights, matrix-valued orthogonal polynomials and Jacobi operators 3.5.1 Matrix weights 3.5.2 Matrix-valued orthogonal polynomials 3.5.3 Link to case of l2(Z) 3.5.4 Reducibility 3.5.5 Exercises 3.6 The J-matrix method 3.6.1 Schr ̈ odinger equation with the Morse potential 3.6.2 A tridiagonal differential operator 3.6.3 J-matrix method with matrix-valued orthog- onal polynomials 3.6.4 Exercises 3.7 Appendix: The spectral theorem 3.7.1 Hilbert spaces and operators 3.7.2 Hilbert C∗-modules 3.7.3 Unbounded operators 3.7.4 The spectral theorem for bounded self- adjoint operators 3.7.5 Unbounded self-adjoint operators 3.7.6 The spectral theorem for unbounded self- adjoint operators 3.8 Hints and answers for selected exercises References 4 Elliptic Hypergeometric Functions 4.1 Elliptic functions 4.1.1 Definitions 4.1.2 Theta functions 4.1.3 Factorization of elliptic functions 4.1.4 The three-term identity 4.1.5 Even elliptic functions 4.1.6 Interpolation and partial fractions 4.1.7 Modularity and elliptic curves 4.1.8 Comparison with classical notation 4.2 Elliptic hypergeometric functions 4.2.1 Three levels of hypergeometry 4.2.2 Elliptic hypergeometric sums 4.2.3 The Frenkel–Turaev sum 4.2.4 Well-poised and very well-poised sums 4.2.5 The sum 12V11 4.2.6 Biorthogonal rational functions 4.2.7 A quadratic summation 4.2.8 An elliptic Minton summation 4.2.9 The elliptic gamma function 4.2.10 Elliptic hypergeometric integrals 4.2.11 Spiridonov’s elliptic beta integral 4.3 Solvable lattice models 4.3.1 Solid-on-solid models 4.3.2 The Yang–Baxter equation 4.3.3 The R-operator 4.3.4 The elliptic SOS model 4.3.5 Fusion and elliptic hypergeometry References 5 Combinatorics of Orthogonal Polynomials and their Moments 5.1 Introduction 5.2 General and combinatorial theories of formal OPS 5.2.1 Formal theory of orthogonal polynomials 5.2.2 The Flajolet–Viennot combinatorial approach 5.3 Combinatorics of generating functions 5.3.1 Exponential formula and Foata’s approach 5.3.2 Models of orthogonal Sheffer polynomials 5.3.3 MacMahon’s Master Theorem and a Mehler- type formula 5.4 Moments of orthogonal Sheffer polynomials 5.4.1 Combinatorics of the moments 5.4.2 Linearization coefficients of Sheffer polynomials 5.5 Combinatorics of some q-polynomials 5.5.1 Al-Salam–Chihara polynomials 5.5.2 Moments of continuous q-Hermite, q-Charlier and q-Laguerre polynomials 5.5.3 Linearization coefficients of continuous q- Hermite, q-Charlier and q-Laguerre polynomials 5.5.4 A curious q-analogue of Hermite polynomials 5.5.5 Combinatorics of continued fractions and γ-positivity 5.6 Some open problems References "On July 11-15, 2016, we organized a summer school, Orthogonal Polynomials and Special Functions, Summer School 6 (OPSF-S6) which was hosted at the University of Maryland, College Park, Maryland. This summer school was co-organized with Kasso Okoudjou, Professor and Associate Chair, Department of Mathematics, and Norbert Wiener Center for Harmonic Analysis and Applications. OPSF-S6 was a National Science Foundation (NSF) supported summer school on orthogonal polynomials and special functions which received partial support from the Institute forMathematics and its Applications (IMA),Minneapolis, Minnesota. Twenty-two, undergraduates, graduate students, and young researchers attended the summer school from the USA, China, Europe, Morocco and Tunisia, hoping to learn a new state of the art in these subject areas. Since 1970, the subjects of special functions and special families of orthogonal polynomials, have gone through major developments. The Wilson and Askey-Wilson polynomials paved the way for a better understanding of the theory of hypergeometric and basic hypergeometric series and shed new light on the pioneering work of Rogers and Ramanujan. This was combined with advances in the applications of q-series in number theory through the theory of partitions and allied subjects. When quantum groups arrived, the functions which appeared in their representation theory turned out to be the same q-functions which were recently developed at that time. This motivated researchers to revisit the old Bochner problem of studying polynomial solutions to second order differential, difference, or q-difference equations, which are of Sturm-Liouville type and have polynomial coefficients"-- Provided by publisher "On July 11-15, 2016, we organized a summer school, Orthogonal Polynomials and Special Functions, Summer School 6 (OPSF-S6) which was hosted at the University of Maryland, College Park, Maryland. This summer school was co-organized with Kasso Okoudjou, Professor and Associate Chair, Department of Mathematics, and Norbert Wiener Center for Harmonic Analysis and Applications. OPSF-S6 was a National Science Foundation (NSF) supported summer school on orthogonal polynomials and special functions which received partial support from the Institute forMathematics and its Applications (IMA),Minneapolis, Minnesota. Twenty-two, undergraduates, graduate students, and young researchers attended the summer school from the USA, China, Europe, Morocco and Tunisia, hoping to learn a new state of the art in these subject areas. Since 1970, the subjects of special functions and special families of orthogonal polynomials, have gone through major developments. The Wilson and Askey-Wilson polynomials paved the way for a better understanding of the theory of hypergeometric and basic hypergeometric series and shed new light on the pioneering work of Rogers and Ramanujan. This was combined with advances in the applications of q-series in number theory through the theory of partitions and allied subjects. When quantum groups arrived, the functions which appeared in their representation theory turned out to be the same q-functions which were recently developed at that time. This motivated researchers to revisit the old Bochner problem of studying polynomial solutions to second order differential, difference, or q-difference equations, which are of Sturm-Liouville type and have polynomial coefficients"-- Résumé de l'éditeur This is a collection of graduate-level introductions to five areas of current research interest in orthogonal polynomials and special functions. It derives from the OPSF-S6 Summer School lectures given by international authorities and has been carefully edited into a coherent whole, with examples and exercises.