The abstract theory of boundary triples is applied to the classical Jacobi differential operator and its powers in order to obtain the Weyl m-function for several self-adjoint extensions with interesting boundary conditions: separated, periodic and those that yield the Friedrichs extension. These matrix-valued Nevanlinna-Herglotz m-functions are, to the best knowledge of the author, the first explicit examples to stem from singular higher-order differential equations. The creation of the boundary triples involves taking pieces, determined in [26], of the principal and non-principal solutions of the differential equation and putting them into the sesquilinear form to yield maps from the maximal domain to the boundary space. These maps act like quasi-derivatives, which are usually not well-defined for all functions in the maximal domain of singular expressions. However, well-defined regularizations of quasi-derivatives are produced by putting the pieces of the non-principal solutions through a modified Gram-Schmidt process.

We set out to build a framework for self-adjoint extension theory for powers of the Jacobi differential operator that does not make use of classical deficiency elements. Instead, we rely on simpler functions that capture the impact of these elements on extensions but are defined by boundary asymptotics. This new perspective makes calculations much more accessible and allows for a more nuanced analysis of the associated domains. The maximal domain for n-th composition of the Jacobi operator is characterized in terms of a smoothness condition for each derivative, and the endpoint behavior of functions in the underlying Hilbert space can then be classified, for j is an element of N-0, by (1 - x)(j), (1 + x)(j), (1 - x)(-alpha+j) and (1 + x)(beta+j). Most of these behaviors can only occur when functions are in the associated minimal domain, and this leads to a formulation of the defect spaces with a convenient basis. Self-adjoint extensions, including the important left-definite domain, are then given in terms of the new basis functions for the defect spaces using GKN theory. Comments are made for the Laguerre operator as well.

In the early 2000s, Littlejohn and Wellman developed so-called nth left-definite theory. Namely, they fully determined the 'left-definite domains' and spectral properties of powers of self-adjoint Sturm-Liouville operators associated with classical orthogonal polynomials. We study how these left-definite domains relate with explicit classical Glazman-Krein-Naimark (GKN) boundary conditions. When n is small, we significantly simplify previously challenging analysis by introducing an explicit method for checking whether a given set of functions yields GKN conditions. This reduces to computing the rank of a relatively small matrix. We include explicit computations for n = 2, ... , 5. Further, for arbitrary powers n of Sturm-Liouville operators with a complete system of orthogonal eigenfunctions, we show that these left-definite domains are given by GKN boundary conditions involving some of the polynomial eigenfunctions. We also study and extend a conjecture by Littlejohn-Wicks regarding the equality of four different formulations for these domains.