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

Intro -- Table of Contents -- List of Figures -- Acknowledgements -- Chapter 1. Introduction -- 1.1 The Vibrating String -- 1.2 Network Theory -- 1.3 Random Walk With Discrete Time Process -- Chapter 2. Regular Sturm-Liouville Problem -- 2.1 Set Up -- 2.2 Preliminary Results -- 2.3 Orthogonality, Eigenfunction Expansion, Spectral Function, and Green's Function -- Chapter 3. Singular Sturm-Liouville Problem -- 3.1 Definition -- 3.2 C[sub(b')] Circles -- 3.3 C[sub(a')] Circles -- 3.4 Existence of Boundary Conditions -- 3.5 Singular Boundary Value Problems -- 3.6 Green's Function -- 3.7 Self-Adjointness -- 3.8 λ-Independence of Boundary Conditions -- 3.9 Green's Formulas -- 3.10 Spectral Resolution -- 3.11 Limit-Point and Limit-Circle Tests -- Chapter 4. Polynomial Solutions -- 4.1 Formal Self-Adjointness -- 4.2 Polynomial Solutions -- 4.3 Orthogonality of Eigenfunctions -- 4.4 Eigenfunction Expansion -- Chapter 5. Polynomial Examples -- 5.1 Classification -- 5.2 Recurrence Relations -- 5.3 Weight Functions and Self-Adjoint Forms -- 5.4 Orthogonality -- 5.5 Evaluation of the ||.||[sup(2)] -- 5.6 Zeros -- Chapter 6. The Four Representative Examples -- 6.1 The Generalized Tchebyshev Polynomials -- 6.2 The Generalized Laguerre Polynomials -- 6.3 The Krawtchouk Polynomials -- 6.4 The Charlier Polynomials -- Chapter 7. Left-Definite Spaces -- 7.1 Finite Intervals -- 7.2 Infinite Intervals -- References.

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