# Second-Order Sturm-Liouville Difference Equations and Orthogonal Polynomials

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*Alouf Jirari*

This well-written book is a timely and significant contribution to the understanding of difference equations. Presenting machinery for analyzing many discrete physical situations, the book will be of interest to physicists and engineers as well as mathematicians. The book develops a theory for regular and singular Sturm-Liouville boundary value problems for difference equations, generalizing many of the known results for differential equations. Discussing the self-adjointness of these problems as well as their abstract spectral resolution in the appropriate \(L^2\) setting, the book gives necessary and sufficient conditions for a second-order difference operator to be self-adjoint and have orthogonal polynomials as eigenfunctions. These polynomials are classified into four categories, each of which is given a properties survey and a representative example. Finally, the book shows that the various difference operators defined for these problems are still self-adjoint when restricted to “energy norms”. This book is suitable as a text for an advanced graduate course on Sturm-Liouville operators or on applied analysis.

#### Table of Contents

# Table of Contents

## Second-Order Sturm-Liouville Difference Equations and Orthogonal Polynomials

- Table of Contents v6 free
- List of Figures viii9 free
- Acknowledgements ix10 free
- Chapter 1. Introduction 112 free
- Chapter 2. Regular Sturm-Liouville Problem 718
- Chapter 3. Singular Sturm-Liouville Problem 2839
- 3.1 Definition 2839
- 3.2 C[sub(b')] Circles 2839
- 3.3 C[sub(a')] Circles 3546
- 3.4 Existence of Boundary Conditions 3849
- 3.5 Singular Boundary Value Problems 4051
- 3.6 Green's Function 4152
- 3.7 Self-Adjointness 4354
- 3.8 λ-Independence of Boundary Conditions 4556
- 3.9 Green's Formulas 5061
- 3.10 Spectral Resolution 5364
- 3.11 Limit-Point and Limit-Circle Tests 7081

- Chapter 4. Polynomial Solutions 7485
- Chapter 5. Polynomial Examples 8596
- Chapter 6. The Four Representative Examples 101112
- Chapter 7. Left-Definite Spaces 124135
- References 137148