Classical and Sobolev orthogonality of the nonclassical Jacobi polynomials with parameters $$\alpha =\beta =-1$$

Abstract

In this paper, we consider the second-order differential expressionThis is the Jacobi differential expression with non-classical parameters α = β = −1 in contrast to the classical case when α, β > −1. For fixed k ≥ 0 and appropriate values of the spectral parameter λ, the equation ℓ[y] = λy has, as in the classical case, a sequence of (Jacobi) polynomial solutions {PThese Jacobi polynomial solutions of degree ≥ 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 ≥ 0 are orthogonal with respect to a Sobolev inner product. Our main result in this paper is to construct a selfadjoint operator T, generated by ℓ[·], 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.Date: May 22, 2012 (BL2v4.tex).

Operational Methods in the Study of Sobolev-Jacobi Polynomials

“…(x) = 0. Referring to [24] for a detailed discussion, the resolution of this problem proposed by Kwon and Littlejohn [28][29][30]32] consists in redefining the notion of orthogonality: one no longer requires orthogonality of the polynomials with respect to the inner product Φ α,β , but instead defines a Sobolev-type inner product. We quote from [30] the relevant material (cf.…”