From Operator Theory to Orthogonal Polynomials, Combinatorics, and Number Theory: A Volume in Honor of Lance Littlejohn's 70th Birthday

Preface
	References
Contents
Compositions and Chebyshev Polynomials
	1 Introduction
	2 Proof of Theorem 1
	3 Proof of Theorem 2
	4 Proof of Theorem 3
	5 Proofs of Theorems 4 and Corollary 1
	6 Proof of Theorem 6 and Corollaries
	7 Further Topics
	References
Non-negative Extensions of Hamiltonian Systems
	1 Introduction
	2 Preliminaries
	3 The Friedrichs Extension TF of T0
	4 Characterisation of Non-negative Extensions TB
	5 Example: A Fourth Order ODE
	References
On Simon's Hausdorff Dimension Conjecture
	1 Introduction
	2 A Weak Version of Simon's Hausdorff Dimension Conjecture
		2.1 A Basic Estimate
		2.2 Prüfer Variables
		2.3 Unboundedness and Infinite Energy
		2.4 Proof of Theorem 1.1 and Corollary 1.2
	References
Hypergeometric Functions over Finite Fields and Modular Forms: A Survey and New Conjectures
	1 Introduction
	2 Preliminaries
	3 Weight Two Newforms
	4 Higher Weight Newforms
		4.1 The Conjectures of Rodriguez Villegas
		4.2 Conjectures of Evans
		4.3 Relations with Ramanujan's τ-Function
		4.4 Other Relations
	5 Trace Formulas for Hecke Operators
	6 New Relations
	References
Ballistic Transport for Periodic Jacobi Operators on Zd
	1 Introduction
	2 Decomposition of J
	3 Ballistic Motion
	References
Perspectives on General Left-Definite Theory
	1 Introduction
		1.1 Notation
	2 Sturm–Liouville Operators
	3 Left-Definite Theory
	4 Comparison with BKV Semi-Bounded Form Theory
	5 Scale of Spaces from Singular Perturbation Theory
	6 Perturbation Setup
	Appendix: Extension Theory
	References
Sampling in the Range of the Analysis Operator of a Continuous Frame Having Unitary Structure
	1 Statement of the Problem
	2 Some Preliminaries
		2.1 Continuous and Discrete Frames
		2.2 Discrete Convolution Systems and Frames of Translates
	3 The Subspace of L2(G) Where the Sampling Is Carried Out
		3.1 Sampling Data as a Filtering Process
	4 The Main Sampling Result and Consequences
		4.1 Sampling at a Subgroup R with Finite Index in H
		4.2 Additional Notes and Remarks
		4.3 The Case of a Semi-Direct Product of Groups
			Euclidean Motion Group and Crystallographic Subgroups
		4.4 Some Final Comments
	References
An Extension of the Coherent Pair of Measures of the Second Kind on the Unit Circle
	1 Introduction
	2 Coherent Pairs of Measures of the Second Kind
		2.1 The Case dμ1(z) = 12πi zdz
		2.2 The Case dμ1(z)=1|z-u|212πi zdz, u≠0
		2.3 A General Case
	3 Hessenberg Matrices
	4 Sobolev OPUC
	References
Bessel-Type Operators and a Refinement of Hardy's Inequality
	1 Introduction
	2 An Exactly Solvable, Strongly Singular, Periodic Schrödinger Operator
	3 A Refinement of Hardy's Inequality
	A.1 The Weyl–Titchmarsh–Kodaira m-Function Associated with Ts,F
	B.1 Remarks on Hardy-Type Inequalities
	References
Spectral Theory of Exceptional Hermite Polynomials
	1 Introduction
	2 Some Spectral Theory
	3 The Formal Theory of Exceptional Hermite Polynomials
		3.1 Multi-Step Factorization Chains
		3.2 The Norm Identity
	4 The L2 Theory
	References
Occupation Time for Classical and Quantum Walks
	1 Introduction
	2 A Look at the Classical Discrete Case
	3 Occupation Times for Quantum Walks
	4 A Look at the Hadamard Walk
	5 The Walk with a Constant Coin
	6 The Even Verblunsky Coefficients Tend to One
	7 A Look at the Riesz Walk
	References
On Foci of Ellipses Inscribed in Cyclic Polygons
	1 Introduction
	2 Background and Notation
	3 The Quadrilateral Case
	4 The Hexagon Case
	5 The Pentagon Case
	References
A Differential Analogue of Favard's Theorem
	1 Introduction
	2 The Main Theory
		2.1 Fundamental Results
		2.2 Relation to Existing Work
	3 Examples
		3.1 Jacobi
		3.2 Hermite
		3.3 Generalized Hermite
		3.4 Laguerre
		3.5 Generalized Laguerre
		3.6 Continuous Hahn
	4 Computational Considerations
		4.1 Computation of Expansion Coefficients
		4.2 Approximation Theory on the Real Line
	5 Periodic Bases Arising from Discrete Orthogonal Polynomials
	6 Challenges and Outlook
		6.1 Transform Pairs
		6.2 Location of Zeros
		6.3 Sobolev Orthogonality
		6.4 Beyond the Canonical Form
		6.5 A Freudian Slip—Why We Need More Polynomials
	References
Intrinsic Properties of Strongly Continuous Fractional Semigroups in Normed Vector Spaces
	1 Introduction
	2 Background
		2.1 Logarithmic Norms on Banach Spaces
		2.2 Logarithmic Norm Bounds of Classical Semigroups
	3 Fractional Semigroups
		3.1 Mittag-Leffler and Wright Functions
		3.2 Logarithmic Norm Bounds of Fractional Semigroups
	4 Conclusions and Future Endeavors
	References
The BFK-gluing Formula for Zeta-determinants and the Conformal Rescaling of a Metric
	1 Introduction
	2 The Metric Rescaling and Invariance Theory
	3 Proof of Theorem 1
	4 Conclusions
	References
New Representations of the Laguerre–Sobolev and Jacobi–Sobolev Orthogonal Polynomials
	1 Introduction
	2 Two Representations of the Laguerre–Sobolev Polynomials
	3 New Representations of the Jacobi–Sobolev Polynomials
	References
Compactness, or Lack Thereof, for the Harmonic Double Layer
	1 Compactness of the Harmonic Double Layer Operator on Lebesgue Spaces
	2 Failure of Compactness for the Harmonic Double Layer Operator
	References
Weighted Chebyshev Polynomials on Compact Subsets of the Complex Plane
	1 Introduction
	2 Existence, Uniqueness, and Characterization of Weighted Chebyshev Polynomials
	3 Bounds for Weighted Chebyshev Polynomials
	References
The Eichler Integral of E2 and q-brackets of t-hook Functions
	1 Introduction and Statement of Results
	2 Nuts and Bolts
		2.1 A Formula of Han
		2.2 A Formula of Berndt
	3 Proofs of Results
	4 Some Examples
	References