Orthogonal Polynomials: 2nd Aims-Volkswagen Stiftung Workshop, Douala, Cameroon, 5-12 October, 2018

Foreword
	Acknowledgements
Contents
Part I Introduction to Orthogonal Polynomials
	An Introduction to Orthogonal Polynomials
		1 Introduction: An Example of a Family of Orthogonal Polynomials
		2 Construction of a System of Orthogonal Polynomials
		3 Basic Properties of Orthogonal Polynomials
			3.1 The Uniqueness of a Family of Orthogonal Polynomials
			3.2 The Matrix Representation
			3.3 The Three-Term Recurrence Relation
			3.4 The Christoffel-Darboux Formula
			3.5 The Interlacing Properties of the Zeros
			3.6 Solution to the L2(α) Extremal Problem
			3.7 Gauss Quadrature Formula
			3.8 Concluding Remarks
		References
	Classical Continuous Orthogonal Polynomials
		1 Definitions
		2 Orthogonality of the Derivatives
		3 Second-Order Differential Equation
		4 Rodrigues\' Formula
		5 Classification of Classical Orthogonal Polynomials of a Continuous Variable
			5.1 Classical Orthogonal Polynomials Obtained if deg(ϕ)=2
			5.2 Classical Orthogonal Polynomials Obtained if deg(ϕ)=1
			5.3 Classical Orthogonal Polynomials Obtained if deg(ϕ)=0
		6 Characterization Theorem of Classical Orthogonal Polynomials
		7 Tutorial
		References
	Generating Functions and Hypergeometric Representations of Classical Continuous Orthogonal Polynomials
		1 Introduction
		2 Generating Functions of Classical Continuous Orthogonal Polynomials
		3 Hypergeometric Representations of Classical Orthogonal Polynomials
		4 Exercises
		References
	Properties and Applications of the Zeros of Classical Continuous Orthogonal Polynomials
		1 Introduction
		2 The Location of Zeros of Orthogonal Polynomials
		3 Gauss Quadrature
		References
	Inversion, Multiplication and Connection Formulae of Classical Continuous Orthogonal Polynomials
		1 The Classical Orthogonal Polynomials of a Continuous Variable and Their Generating Functions
		2 Inversion Problem Using Generating Functions
		3 Multiplication Formula Using Generating Functions (See e.g. Chaggarakoepf2007, Rainville1960, Tcheutiaetall2016)
		4 Connection Formula Using Generating Functions(See e.g. Rainville1960)
		5 Structure Relations and Applications
		References
	Classical Orthogonal Polynomials of a Discrete and a q-Discrete Variable
		1 Introduction
		2 Classical Orthogonal Polynomials of a Discrete Variable
			2.1 Definitions and Preliminary Results
			2.2 General Polynomial Solutions of the Hypergeometric Discrete Difference Equation
			2.3 The Four Classical Discrete Orthogonal Polynomials
				2.3.1 Charlier Polynomials
				2.3.2 Meixner Polynomials
				2.3.3 Kravchuk Polynomials
				2.3.4 Hahn Polynomials
			2.4 Some Structure Formulas for the Charlier Polynomials
				2.4.1 The Inversion Formula
				2.4.2 A Connection Formula
				2.4.3 An Addition Formula
		3 Classical Orthogonal Polynomials of a q-Discrete Variable
			3.1 Definitions and Preliminary Results
			3.2 Polynomial Solutions of the q-Difference Equation
			3.3 Some Structure Formulas for the Al-Salam Carlitz I Polynomials
				3.3.1 The Inversion Formula
				3.3.2 A Connection Formula
				3.3.3 A q-Addition Formula
				3.3.4 A Multiplication Formula
		References
	Computer Algebra, Power Series and Summation
		1 Introduction
		2 Taylor Polynomials
		3 Holonomic Power Series
		4 Fasenmyer\'s Algorithm
		5 Gosper\'s Algorithm
		6 Zeilberger\'s Algorithm
		7 CAOP
		8 Petkovšek\'s and van Hoeij\'s Algorithm
		9 Hypergeometric Identities
		10 Generating Functions
		11 Almkvist–Zeilberger Algorithm
		12 Basic Hypergeometric Series
		References
	On the Solutions of Holonomic Third-Order Linear Irreducible Differential Equations in Terms of Hypergeometric Functions
		1 Introduction
		2 Previous Works
			2.1 First-Order Holonomic Differential Equations
			2.2 Second-Order Holonomic Differential Equations
				2.2.1 L Is Reducible
				2.2.2 L Is Irreducible
				2.2.3 Generalized Hypergeometric Functions
			2.3 Third-Order Holonomic Differential Equations
				2.3.1 L Is Reducible
				2.3.2 L Is Irreducible
		3 My Work
			3.1 Our Main Objective
			3.2 The Method
			3.3 Our Inputs
			3.4 Steps of the Resolution
			3.5 Singularities
			3.6 Generalized Exponents
				3.6.1 Generalized Exponents and Singularities
				3.6.2 Generalized Exponents and Exp-Product Transformation
				3.6.3 Generalized Exponents and Gauge Transformation
				3.6.4 Generalized Exponents and Change of Variable Transformation
			3.7 How to Find the Transformation Parameter(s)
				3.7.1 How to Find the Change of Variable Parameter f
				3.7.2 How to Find the Parameter(s) of Our Chosen Special Function S{1F12,  0F2,  1F2,  2F2}
				3.7.3 How to Find the Exp-Product Parameter r
				3.7.4 How to Find the Gauge Parameters r0,r1 and r2
				3.7.5 Example
		4 Conclusion
		References
	The Gamma Function
		1 Definition
		2 Properties of the Gamma and Beta Functions
		3 The Beta Function
		References
Part II Recent Research Topics in Orthogonal Polynomials and Applications
	Hypergeometric Multivariate Orthogonal Polynomials
		1 Classical Univariate Case: From Hermite to q-Racah Polynomials
		2 Hypergeometric Multivariate Orthogonal Polynomials
			2.1 Bivariate Orthogonal Polynomials: Continuous, Discrete and q-Analogues
			2.2 Bivariate Orthogonal Polynomials on Nonuniform Lattices
		References
	Signal Processing, Orthogonal Polynomials, and Heun Equations
		1 Introduction
		2 Motivation and Background
			2.1 Time and Band Limiting
			2.2 The Heun Operator
		3 The Askey Scheme and Bispectral Problems
			3.1 An Algebraic Description
			3.2 Duality
		4 Tridiagonalization of the Hypergeometric Operator
			4.1 The Wilson and Racah Polynomials from the Jacobi Polynomials
		5 The Algebraic Heun Operator
			5.1 A Discrete Analog of the Heun Operator
			5.2 The Algebraic Heun Operator of the Hahn Type
		6 Application to Time and Band Limiting
		7 Conclusion
		References
	Some Characterization Problems Related to Sheffer Polynomial Sets
		1 Introduction
		2 Preliminary Results
			2.1 Operators
		3 Properties of Sheffer Polynomials
		4 Characterization Problems for Sheffer Sets
			4.1 Characterization Theorem
			4.2 Examples
			4.3 Counting d-OPSs of Sheffer Type
			4.4 Classification of d-OPSs of Sheffer Type
				4.4.1 Case d=1
				4.4.2 Case d=2
				4.4.3 Case d=3
				4.4.4 (d+1)-Fold Symmetric d-OPS of Sheffer Type
		References
	From Standard Orthogonal Polynomials to Sobolev Orthogonal Polynomials: The Role of Semiclassical Linear Functionals
		1 Introduction
		2 Background
		3 Discrete Darboux Transformations
			3.1 Christoffel Transformation
			3.2 Geronimus Transformation
			3.3 Uvarov Transformation
		4 Semiclassical Linear Functionals
		5 Examples of Semiclassical Orthogonal Polynomials
		6 Analytic Properties of Orthogonal Polynomials in Sobolev Spaces
			6.1 Coherent Pairs of Measures and Sobolev Orthogonal Polynomials
				6.1.1 Generalized Coherent Pairs
			6.2 Sobolev-Type Orthogonal Polynomials
			6.3 Asymptotics of Sobolev Orthogonal Polynomials
				6.3.1 Continuous Sobolev Inner Products
		7 Sobolev Orthogonal Polynomials of Several Variables
			7.1 Orthogonal Polynomials of Several Variables
				7.1.1 Sobolev Orthogonal Polynomials on the Unit Ball
				7.1.2 Sobolev Orthogonal Polynomials on Product Domains
		References
	Two Variable Orthogonal Polynomials and Fejér-Riesz Factorization
		1 Introduction
		2 Scalar Orthogonal Polynomials on the Unit Circle
		3 Matrix Orthogonal Polynomials on the Unit Circle
		4 Orthogonal Polynomials on the Bicircle
		References
	Exceptional Orthogonal Polynomials and Rational Solutions to Painlevé Equations
		1 Introduction
		2 Darboux Transformations
			2.1 Exact Solvability by Polynomials
			2.2 Schrödinger and Algebraic Operators
			2.3 Rational Darboux Transformations
			2.4 Iterated or Darboux-Crum Transformations
		3 The Bochner Problem: Classical and Exceptional Polynomials
			3.1 Sturm-Liouville Problems
			3.2 Classical Orthogonal Polynomials
			3.3 Exceptional Polynomials and Operators
		4 Symmetric Painlevé Equations and Darboux Dressing Chains
			4.1 Darboux Dressing Chains
		5 Rational Extensions of the Harmonic Oscillator
			5.1 Maya Diagrams
			5.2 Hermite Pseudo-Wronskians
			5.3 Rational Extensions of the Harmonic Oscillator
			5.4 Cyclic Maya Diagrams
		6 Classification of Cyclic Maya Diagrams
			6.1 Indexing Maya p-Cycles
			6.2 Rational Solutions of A4-Painlevé
		References
	(R, p,q)-Rogers–Szegö and Hermite Polynomials, and Induced Deformed Quantum Algebras
		1 Introduction
		2 Preliminaries
			2.1 Hermite Polynomials
			2.2 The Rogers–Szegö Polynomials
			2.3 Second Order q-Differential Equation for Rogers–Szegö Polynomials
		3 (R,p,q)-Generalized Rogers–Szegö Polynomials and Quantum Algebras
		4 Particulars Cases
			4.1 (p,q)-Rogers–Szegö Polynomials and (p,q)-Oscillator from Jagannathan-Srinivasa Deformation JaganSridh
			4.2 Rogers–Szegö Polynomial Associated to the Chakrabarty-Jagannathan Deformation ChakJagan
			4.3 (p,q)-Rogers–Szegö Polynomials Associated with the Quesne\'s Deformed Quantum Algebra HmNe
			4.4 (p,q,μ,ν,h)-Rogers–Szegö Polynomials Associated to Hounkonnou-Ngompe Quantum Algebra HmNe2
		5 Continuous (R, p,q)-Hermite Polynomials
			5.1 Generalization
			5.2 Particular Cases
				5.2.1 Continuous (p,q)-Hermite Polynomials
				5.2.2 Continuous (p-1,q)-Hermite Polynomials
				5.2.3 Continuous (p,q)-Hermite Polynomials Related to (p, q)-Generalization of Quesne Deformation HmNe
				5.2.4 Continuous (p,q,μ,ν,h)-Hermite Polynomials Related to the Hounkonnou-Ngompe Deformation HmNe2
		6 Concluding Remarks
		References
	Zeros of Orthogonal Polynomials
		1 Zeros as Eigenvalues
		2 Monotonicity of the Zeros
		3 Interlacing of Zeros from Different Sequences
			3.1 Jacobi Polynomials
			3.2 A One-Dimensional Electrostatic Model for Zeros of Different Jacobi Polynomials
		4 Stieltjes Interlacing of Zeros
		5 Bounds for the Zeros of Orthogonal Polynomials
			5.1 Bounds for Zeros from Stieltjes Interlacing
		6 Distance Between the Consecutive Zeros
		References
	Properties of Certain Classes of Semiclassical Orthogonal Polynomials
		1 Introduction
		2 Semiclassical Orthogonal Polynomials
			2.1 Semiclassical Laguerre Polynomials
			2.2 Generalized Freud Polynomials
		3 A Characterisation of Askey–Wilson Polynomials
		References
	Orthogonal Polynomials and Computer Algebra
		1 Orthogonal Polynomials
		2 Classical Orthogonal Polynomials
		3 Classical Discrete Orthogonal Polynomials
		4 Classical q-Orthogonal Polynomials and the Askey–Wilson Scheme
		5 Computer Algebra Applied to Classical Orthogonal Polynomials
		6 Epilogue
		References
	Spin Chains, Graphs and State Revival
		1 Introduction
		2 Fractional Revival (FR) and Perfect State Transfer (PST) in a One Dimensional Spin Chain
		3 Para-Krawtchouk and Krawtchouk Models
		4 Quantum Walk on the Hypercube
			4.1 A Brief Review of the Hamming Scheme
			4.2 Projection of the Quantum Walk on the Hypercube to the Krawtchouk Model
		5 Bivariate Krawtchouk Polynomials
			5.1 Algebraic Interpretation: SO(3)
			5.2 Relationship to Generalized Hamming Scheme
		6 FR and PST in Two-Dimensional Spin Lattices
		7 Concluding Remarks
		References
	An Introduction to Special Functions with Some Applications to Quantum Mechanics
		1 An Introduction to Special Functions
			1.1 Classical Hypergeometric Functions
				1.1.1 Method of Undetermined Coefficients
				1.1.2 Some Solutions of Hypergeometric Equations
			1.2 Integral Representations
				1.2.1 Transformation to the Simplest Form
				1.2.2 Main Theorem
				1.2.3 Integrals for Hypergeometric and Bessel Functions
			1.3 Classical Orthogonal Polynomials
				1.3.1 Main Property
				1.3.2 Rodrigues Formula
				1.3.3 Orthogonality
				1.3.4 Classification
				1.3.5 Functions of the Second Kind
				1.3.6 Complex Orthogonality
		Exercises
		2 Some Problems of Nonrelativistic and Relativistic Quantum Mechanics
			2.1 Generalized Equation of Hypergeometric Type
			2.2 Classical Orthogonal Polynomials and Eigenvalue Problems
				2.2.1 Example: Linear Harmonic Oscillator
			2.3 Method of Separation of Variables and Its Extension
				2.3.1 Method of Separation of Variables
				2.3.2 Dirac-Type Systems
			2.4 Nonrelativistic Coulomb Problem
				2.4.1 Radial Equation
				2.4.2 Quantization
				2.4.3 Summary: Wave Functions and Energy Levels
			2.5 Matrix Elements
				2.5.1 General Results
				2.5.2 Special Cases
		3 Relativistic Coulomb Problem
			3.1 Dirac Equation
			3.2 Relativistic Coulomb Wave Functions and Discrete Energy Levels
			3.3 Solution of Dirac Wave Equation for Coulomb Potential
				3.3.1 The Spinor Spherical Harmonics
				3.3.2 Separation of Variables in Spherical Coordinates
				3.3.3 Solution of Radial Equations
				3.3.4 Nonrelativistic Limit of the Wave Functions
		4 Symmetry of Quantum Harmonic Oscillators
			4.1 Symmetry and ``Hidden\'\' Solutions
			4.2 Computer Animations
			4.3 The Momentum Representation
			4.4 The Schrödinger Group for Simple Harmonic Oscillators
			4.5 A Complex Parametrization of the Schrödinger Group
			4.6 Discussion
		5 Expectation Values in Relativistic Coulomb Problems
			5.1 Evaluation of the Matrix Elements
			5.2 Inversion Formulas
			5.3 Recurrence Relations
			5.4 Special Expectation Values and Their Applications
			5.5 Three-Term Recurrence Relations and Computer Algebra Methods
		Appendix A: Evaluation of an Integral
		Appendix B: Hypergeometric Series, Discrete Orthogonal Polynomials, and Useful Relations
		Appendix C: Dirac Matrices and Inner Product
		References
	Orthogonal and Multiple Orthogonal Polynomials, Random Matrices, and Painlevé Equations
		1 Introduction
		2 Orthogonal Polynomials and Random Matrices
			2.1 Point Processes
			2.2 Determinantal Point Process
			2.3 Random Matrices
			2.4 Random Matrix Ensembles
		3 Multiple Orthogonal Polynomials
			3.1 Special Systems
			3.2 Nearest Neighbor Recurrence Relations
			3.3 Christoffel-Darboux Formula
			3.4 Hermite-Padé Approximation
			3.5 Multiple Hermite Polynomials
				3.5.1 Random Matrices
				3.5.2 Non-intersecting Brownian Motions
			3.6 Multiple Laguerre Polynomials
				3.6.1 Multiple Laguerre Polynomials of the First Kind
				3.6.2 Multiple Laguerre Polynomials of the Second Kind
				3.6.3 Random Matrices: Wishart Ensemble
			3.7 Jacobi-Piñeiro Polynomials
		4 Orthogonal Polynomials and Painlevé Equations
			4.1 Compatibility and Lax Pairs
			4.2 Discrete Painlevé I
			4.3 Langmuir Lattice and Painlevé IV
			4.4 Singularity Confinement
			4.5 Generalized Charlier Polynomials
			4.6 Discrete Painlevé II
			4.7 The Ablowitz-Ladik Lattice and Painlevé III
			4.8 Some More Examples
				4.8.1 Generalized Meixner Polynomials
				4.8.2 Modified Laguerre Polynomials
				4.8.3 Modified Jacobi Polynomials
				4.8.4 q-Orthogonal Polynomials
			4.9 Wronskians and Special Function Solutions
		References