Lectures on Orthogonal Polynomials and Special Functions

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