A spectral study of the second-order exceptional X1-Jacobi differential expression and a related non-classical Jacobi differential expression

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 {Pˆn(α,β)}n=1∞ 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 L2((−1,1);wˆα,β), where wˆα,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 L2((−1,1);wˆα,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 {Pn(−2,β)}n=2∞. We study the corresponding Jacobi differential expression in several Hilbert spaces, including their natural L2 setting and a certain Sobolev space S where the full sequence {Pn(−2,β)}n=0∞ is studied and a careful spectral analysis of the Jacobi expression is carried out.