On Sobolev orthogonal polynomials with coherent pairs The Jacobi case

Abstract

When we investigate the asymptotic properties of orthogonal polynomials with Sobolev inner product f,g = ∫ a b f(x)g(x)dμ(x) + λ ∫ a b f′(x)g′(x) dv(x) we need to know the relations between P n and Q n where P n ( x) and Q n ( x) are the nth monic orthogonal polynomials with respect to d μ and d v, respectively. The pair d μ, d v is called a coherent pair if there exist nonzero constant D n such that Q n(x) = P′ n+1(x) n+1 + D n 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 Q n(x) P n(x) under the Jacobi case. We prove that lim n→∞ D n exists and calculate the limit as well for the Jacobi case.