Zeros of Sobolev orthogonal polynomials of Hermite type

Abstract

Let { S n } n denote a set of polynomials orthogonal with respect to the Sobolev inner product 〈f,g〉 S =∫f(x)g(x) dψ 0(x)+λ∫f ′(x)g ′(x) dψ 1, where λ>0 and {d ψ 0,d ψ 1} is a so-called symmetrically coherent pair with d ψ 0 or d ψ 1 the Hermite measure e −x 2 dx . If d ψ 1 is the Hermite measure, then S n has n different, real zeros. If d ψ 0 is the Hermite measure, then S n has at least n−2 different, real zeros. We determine conditions for S n to have complex zeros.