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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, asequence of (Jacobi) polynomial solutions {P_{n}^{(-1,-1)}}_{n=0}^{\infty}.These Jacobi polynomial solutions of degree \geq 2 form a complete orthogonalset in the Hilbert space L^2((-1,1);(1-x^2)^(-1). Unlike the classicalsituation, every polynomial of degree one is a solution of this eigenvalueequation. Kwon and Littlejohn showed that, by careful selection of this firstdegree solution, the set of polynomial solutions of degree \geq 0 areorthogonal with respect to a Sobolev inner product. Our main result in thispaper is to construct a self-adjoint operator T, generated by \ell[\cdot], inthis Sobolev space that has these Jacobi polynomials as a complete orthogonalset of eigenfunctions. The classical Glazman-Krein-Naimark theory is essentialin helping to construct T in this Sobolev space as is the left-definite theorydeveloped by Littlejohn and Wellman.

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