Nonclassical Jacobi Polynomials and Sobolev Orthogonality

In this paper, we consider the second-order Jacobi differential expression here, the Jacobi parameters are alpha > -1 and beta = -1. This is a nonclassical setting since the classical setting for this expression is generally considered when alpha, beta > -1. In the classical setting, it is well-known that the Jacobi polynomials are (orthogonal) eigenfunctions of a self-adjoint operator T (alpha, beta) , generated by the Jacobi differential expression, in the Hilbert space L (2)((-1,1);(1-x) (alpha) (1 + x) (beta)). When alpha > -1 and beta = -1, the Jacobi polynomial of degree 0 does not belong to the Hilbert space L (2)((-1,1);(1 - x) (alpha) (1 + x)(-1)). However, in this paper, we show that the full sequence of Jacobi polynomials forms a complete orthogonal set in a Hilbert-Sobolev space W (alpha) , generated by the inner product We also construct a self-adjoint operator T (alpha) , generated by l (alpha,-1)[center dot] in W (alpha) , that has the Jacobi polynomials as eigenfunctions.