Nonclassical Jacobi Polynomials and Sobolev Orthogonality

Abstract

In this paper, we consider the second-order Jacobi differential expression $$\ell_{\alpha,\beta}[y](x)=\dfrac{-1}{(1-x)^{a}(1+x)^{-1}}\left( (1-x\right) ^{\alpha+1}y^{\prime}(x))^{\prime} \quad(x\in(-1,1));$$ here, the Jacobi parameters are α > −1 and β = −1. This is a nonclassical setting since the classical setting for this expression is generally considered when α, β > −1. In the classical setting, it is well-known that the Jacobi polynomials \({\{P_{n}^{(\alpha,\beta)}\}_{n=0}^{\infty}}\) are (orthogonal) eigenfunctions of a self-adjoint operator Tα, β, generated by the Jacobi differential expression, in the Hilbert space L2((−1,1);(1−x)α(1 + x)β). When α > −1 and β = −1, the Jacobi polynomial of degree 0 does not belong to the Hilbert space L2((−1,1);(1 − x)α(1 + x)−1). However, in this paper, we show that the full sequence of Jacobi polynomials \({\{P_{n} ^{(\alpha,-1)}\}_{n=0}^{\infty}}\) forms a complete orthogonal set in a Hilbert–Sobolev space Wα, generated by the inner product $$\phi\left( f,g\right) :=f(-1)\overline{g}(-1)+\int\limits_{-1}^{1}f^{\prime }(t)\overline{g}^{\prime}(t)(1-t)^{\alpha+1}dt.$$ We also construct a self-adjoint operator Tα, generated by lα,−1[·] in Wα, that has the Jacobi polynomials \({\{P_{n}^{(\alpha,-1)}\}_{n=0}^{\infty}}\) as eigenfunctions.