Applied left-definite theory: The Jacobi polynomials, their Sobolev orthogonality, and self-adjoint operators

It is well known that, for --alpha, --beta, --alpha, --beta -- 1 &nisin; N the Jacobi polynomials Pab nx infinityn=0 are orthogonal on R with respect to a bilinear form of the type f,gm =Rfgdm, for some measure mu. However, for negative integer parameters alpha and beta, an application of Favard's theorem shows that the Jacobi polynomials cannot be orthogonal on the real line with respect to a bilinear form of this type for any positive or signed measure. But it is known that they are orthogonal with respect to a Sobolev inner product. In this work, we first consider the special case where alpha = beta = --1. We shall discuss the Sobolev orthogonality of the Jacobi polynomials and construct a self-adjoint operator in a certain Hilbert-Sobolev space having the entire sequence of Jacobi polynomials as eigenfunctions. The key to this construction is the left-definite theory associated with the Jacobi differential equation, and the left-definite spaces and operators will be constructed explicitly. The results will then be generalized to the case where alpha > --1, beta = --1.