On quasi-orthogonal polynomials

Abstract

In the course of a proof, A. Rosenthal [2] used a class of polynomials which, upon specialization of certain constants, may be defined bydxn These polynomials can be expressed in terms of the Hermite polynomials as follows:From this relation and the well-known orthogonality of the Hermite polynomials, it follows that these polynomials satisfy the "quasiorthogonal ity" relations-COThe preceding admits of an obvious generalization.We will consider real polynomials although, with appropriate notational changes, complex variables could be considered.Let pn(x) denote the reth orthonormal polynomial associated with a distribution da(x) on an interval (a, b).Let k and r be fixed integers,