Differential orthogonality: Laguerre and Hermite cases with applications

Abstract

Let μ be a finite positive Borel measure supported on R, L[f]=xf″+(α+1−x)f′ with α>−1, or L[f]=12f″−xf′, and m a natural number. We study algebraic, analytic and asymptotic properties of the sequence of monic polynomials {Qn}n>m that satisfy the orthogonality relations ∫L[Qn](x)xkdμ(x)=0for all0≤k≤n−1. We also provide a fluid dynamics model for the zeros of these polynomials.