On Sobolev orthogonal polynomials with coherent pairs The Jacobi case

When we investigate the asymptotic properties of orthogonal polynomials with Sobolev inner product f,g = ∫ab f(x)g(x)dμ(x) + λ ∫ab f′(x)g′(x) dv(x) we need to know the relations between Pn and Qn where Pn(x) and Qn(x) are the nth monic orthogonal polynomials with respect to dμ and dv, respectively. The pair dμ, dv is called a coherent pair if there exist nonzero constant Dn such that Qn(x) = P′n+1(x)n+1 + Dn P′(x)n, n⩽1. One can divide the coherent pairs into two cases: the Jacobi case and the Laguerre case. We consider the limit of Qn(x)Pn(x) under the Jacobi case. We prove that limn→∞ Dn exists and calculate the limit as well for the Jacobi case.