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 T _{α, β} , generated by the Jacobi differential expression, in the Hilbert space L ²((−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 L ²((−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 W _α , 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 T _α , generated by ℓ _α,−1[·] in W _α , that has the Jacobi polynomials ${\{P_{n}^{(\alpha,-1)}\}_{n=0}^{\infty}}$ as eigenfunctions.