Nonclassical Jacobi polynomials and Sobolev orthogonality (Q1930330)

The authors consider the nonclassical Jacobi polynomials with parameters \(a>-1\) and \(b=-1\). They prove that the sequence of Jacobi polynomials \(\{{P_{n}^{(a,-1)}}\}_{n=0}^{\infty}\) is complete in the Hilbert-Sobolev space \(W_{a}\) with the inner product \[ \phi (f,g)=f(-1){\bar{g}}(-1)+\int_{-1}^{1}f^{\prime}(t){\bar{g}}^{\prime}(t)(1-t)^{a+1}\,dt. \] They also construct a self-adjoint positively bounded from below operator \(T_{a}\) in \(W_{a}\) having the Jacobi polynomials \(\{{P_{n}^{(a,-1)}}\}_{n=0}^{\infty}\) as a complete set of eigenfunctions.