Differential operators having symmetric orthogonal polynomials as eigenfunctions

摘要

Let the polynomials Pn(x)n=0∞, orthogonal with respect to a symmetric positive definite moment functional σ, be eigenfunctions of a linear differential operator L. We consider the orthogonal polynomials Pnμ(x)n=0∞ and Pnμ,v(x)n=0∞, which are obtained by adding one resp. two symmetric (Sobolev type) terms to σ. In all the cases we 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 L+μA resp. L+μA+vB+μvC. Further it is investigated to what extend the eigenvalues can be chosen arbitrarily and finally expressions are given for the other eigenvalues.