Classical and Sobolev Orthogonality of the Nonclassical Jacobi Polynomials with Parameters {\alpha}={\beta}=-1

Abstract

In this paper, we consider the second-order differential expression \ell [y](x)=(1-x^2)(-(y'(x))'+k(1-x^2)^(-1)y(x))(x\in(-1,1)). This is the Jacobi differential expression with non-classical parameters {\alpha} = {\beta}= -1 in contrast to the classical case when {\alpha}, {\beta} > -1. For fixed k \geq 0 and appropriate values of the spectral parameter {\lambda}, the equation \ell[y]={\lambda}y has, as in the classical case, a sequence of (Jacobi) polynomial solutions {P_{n}^{(-1,-1)}}_{n=0}^{\infty}. These Jacobi polynomial solutions of degree \geq 2 form a complete orthogonal set in the Hilbert space L^2((-1,1);(1-x^2)^(-1). Unlike the classical situation, every polynomial of degree one is a solution of this eigenvalue equation. Kwon and Littlejohn showed that, by careful selection of this first degree solution, the set of polynomial solutions of degree \geq 0 are orthogonal with respect to a Sobolev inner product. Our main result in this paper is to construct a self-adjoint operator T, generated by \ell[\cdot], in this Sobolev space that has these Jacobi polynomials as a complete orthogonal set of eigenfunctions. The classical Glazman-Krein-Naimark theory is essential in helping to construct T in this Sobolev space as is the left-definite theory developed by Littlejohn and Wellman.