Nondiagonal Hermite-Sobolev orthogonal polynomials
Sequences of polynomials \(\{Q_n\}_{n \geq 0}\) orthogonal with respect to the inner product \[ (p,q)_S=\int_{\mathbb R} (p, p') \begin{pmatrix} 1 & \mu \\ \mu & \lambda \end{pmatrix} \begin{pmatrix} q \\ q' \end{pmatrix} e^{-x^2}dx, \quad \lambda -\mu^2>0, \mu \neq 0, \] are studied. First, a relation connecting \(\{Q_n\}\) with Hermite polynomials \(\{H_n\}\) is established, which allows to express the Sobolev norms of \(\{Q_n\}\) in terms of the \(L^2\)-norms of \(\{H_n\}\). These two results lead to a recursive scheme of computation of \(Q_n\) and to a comparative asymptotics of these polynomials with respect to \(H_n\) in \(\mathbb{C} \setminus \mathbb{R}\). Additionally, a differential operator symmetric with respect to \(( \cdot , \cdot)_S\) is computed, yielding a differential equation for \(Q_n\)'s.
scientific article; zbMATH DE number 1546748