It is the aim of this thesis to investigate singular second-order boundary value
problems involving the difference equation
V[p(n)Aj/(n)] + q(n)y(n) = Xw(n)y(n) .
Chapter 1 gives some account of problems with a rather physical character,
described by this difference equation.
Chapter 2 presents a discussion of the Sturm-Liouville boundary value prob-
lem on an interval (a, b) where both a and b are "regular" points.
Chapter 3 derives self-adjoint difference operators when a and/or 6 are "singu-
lar" points. It also takes up the abstract spectral resolution for such operators.
Chapter 4 provides necessary and sufficient conditions for a second-order
difference operator to be formally self-adjoint and have orthogonal polynomi-
als as eigenfunctions.
Chapter 5 shows that these sets of polynomials fall into four categories. It
also surveys their properties, which are familiar in the context of orthogonal
These four classes of polynomials are then illustrated in Chapter 6 by four
representative examples: the generalized Tchebyshev polynomials, the general-
Laguerre polynomials, the Krawtchouk polynomials, and the Charlier polyno-
The closing Chapter 7 is devoted to showing that these polynomials are as-
sociated with difference operators which are self-adjoint in a left-definite setting
V l l