Differential operators having symmetric orthogonal polynomials as eigenfunctions
From authors' abstract: ``Let the polynomials \(\{P_n(x)\}^\infty_{n=0}\), orthogonal with respect to a symmetric positive definite moment functional \(\sigma\), be eigenfunctions of a linear differential operator \({\mathbf L}\). The authors consider the orthogonal polynomials \(\{P^\mu_n(x)\}^\infty_{n= 0}\) and \(\{P^{\mu,\nu}_n(x)\}^\infty_{n= 0}\), which are obtained by adding one resp. two symmetric (Sobolev type) terms to \(\sigma\). In all the cases they derive a representation for the polynomials and show that they are eigenfunctions of one or more linear differential operators (mostly of infinite order) of the form \({\mathbf L}+\mu{\mathbf A}\) resp. \({\mathbf L}+\mu{\mathbf A}+\nu{\mathbf B}+\mu\nu{\mathbf C}\). Further, it is investigated to what extend the eigenvalues can be chosen arbitrarily and finally expressions are given for the other eigenvalues''.
linear differential operators of infinite order
scientific article; zbMATH DE number 1326541