Zeros of Sobolev orthogonal polynomials following from coherent pairs

Abstract

Let { S n λ } denote the monic orthogonal polynomial sequence with respect to the Sobolev inner product 〈f,g〉 S= ∫ −∞ ∞ fg dψ 0+λ ∫ −∞ ∞ f′g′ dψ 1, where {d ψ 0,d ψ 1} is a so-called coherent pair and λ>0. Then S n λ has n different, real zeros. The position of these zeros with respect to the zeros of other orthogonal polynomials (in particular Laguerre and Jacobi polynomials) is investigated. Coherent pairs are found where the zeros of S n−1 λ separate the zeros of S n λ .