Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6418077 | Journal of Mathematical Analysis and Applications | 2015 | 28 Pages |
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 {PËn(α,β)}n=1â 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((â1,1);wËα,β), where wËα,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((â1,1);wËα,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(â2,β)}n=2â. 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(â2,β)}n=0â is studied and a careful spectral analysis of the Jacobi expression is carried out.