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 $$\begin{aligned} \ell [y](x)=(1-x^{2})(-(y^{\prime }(x))^{\prime }+k(1-x^{2})^{-1} y(x))\quad (x\in (-1,1)). \end{aligned}$$ This is the Jacobi differential expression with nonclassical parameters $$\alpha =\beta =-1$$ in contrast to the classical case when $$\alpha ,\beta >-1$$ . For fixed $$k\ge 0$$ 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 $$\ge 2$$ 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 $$\ge 0$$ 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.