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A Spectral Study of the Second-Order Exceptional $X_1$-Jacobi Differential Expression and a Related Non-classical Jacobi Differential Expression

The exceptional $X_{1}$-Jacobi differential expression is a second-order ordinary differential expression with rational coefficients; it was discovered by Gómez-Ullate, Kamran and Milson in 2009. In their work, they showed that there is a sequence of polynomial eigenfunctions $\left\{\widehat{P} _{n}^{(α,β)}\right\}_{n=1}^{\infty}$ called the exceptional $X_{1}$-Jacobi polynomials. There is no exceptional $X_{1}$-Jacobi polynomial of degree zero. These polynomials form a complete orthogonal set in the weighted Hilbert space $L^{2}((-1,1);\widehat{w}_{α,β}),$ where $\widehat{w}_{α,β}$ is a positive rational weight function related to the classical Jacobi weight. Among other conditions placed on the parameters $α$ and $β,$ it is required that $α,β>0.$ In this paper, we develop the spectral theory of this expression in $L^{2}((-1,1);\widehat{w}_{α,β})$. We also consider the spectral analysis of the `extreme' non-exceptional case, namely when $α=0$. In this case, the polynomial solutions are the non-classical Jacobi polynomials $\left\{ P_{n}^{(-2,β)}\right\} _{n=2}^{\infty}.$ We study the corresponding Jacobi differential expression in several Hilbert spaces, including their natural $L^{2}$ setting and a certain Sobolev space $S$ where the full sequence $\left\{ P_{n}^{(-2,β)}\right\} _{n=0}^{\infty}$ is studied and a careful spectral analysis of the Jacobi expression is carried out.