Differential operators having symmetric orthogonal polynomials as eigenfunctions

Let the polynomials {P-n(x)}(n=0)(infinity), orthogonal with respect to a symmetric positive definite moment functional a, be eigenfunctions of a linear differential operator L. We consider the orthogonal polynomials {P-n(mu)(x)}n=0 infinity and {P-n(mu v)(x)}(n=0)(infinity), which are obtained by adding one resp. two symmetric (Sobolev type) terms to a. 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-mu A resp. L + mu A + nu B + mu nu C. Further it is investigated to what extend the eigenvalues can be chosen arbitrarily and finally expressions are given for the other eigenvalues. (C) 1999 Elsevier Science B.V. All rights reserved.