Abstract. The exceptional X1-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 P (α,β)called the exceptional X1-Jacobi polynomials. There is no exceptional X1-Jacobi polynomial of degree zero. These polynomials form a complete orthogonal set in the weighted Hilbert space L 2 ((−1, 1); w α,β ), where 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); 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 P (−2,β). 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 P (−2,β) n ∞ n=0 is studied and a careful spectral analysis of the Jacobi expression is carried out.
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