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We offer a detailed treatment of spectral and Weyl-Titchmarsh-Kodaira theoryfor all self-adjoint Jacobi operator realizations of the differentialexpression \begin{align*} \tau_{\alpha,\beta} = - (1-x)^{-\alpha}(1+x)^{-\beta}(d/dx) \big((1-x)^{\alpha+1}(1+x)^{\beta+1}\big) (d/dx),& \\\alpha, \beta \in \mathbb{R}, \; x \in (-1,1),& \end{align*} in$L^2\big((-1,1); (1-x)^{\alpha} (1+x)^{\beta} dx\big)$, $\alpha, \beta \in\mathbb{R}$. In addition to discussing the separated boundary conditions thatlead to Jacobi orthogonal polynomials as eigenfunctions in detail, weexhaustively treat the case of coupled boundary conditions and illustrate thelatter with the help of the general $\eta$-periodic and Krein--von Neumannextensions. In particular, we treat all underlying Weyl-Titchmarsh-Kodaira andGreen's function induced $m$-functions and revisit their Nevanlinna-Herglotzproperty. We also consider connections to other differential operatorsassociated with orthogonal polynomials such as Laguerre, Gegenbauer, andChebyshev.

The Jacobi operator on $(-1,1)$ and its various $m$-functions