Linear perturbations of differential or difference operators with polynomials as eigenfunctions (Q676004)

Let the polynomials \(\{P_n(x)\}^\infty_{n=0}\) with \(\deg P_n=n\) be eigenfunctions of a linear differential or difference operator \(L\) with eigenvalues \(\{\lambda_n\}^\infty_{n=0}\). In this paper, necessary and sufficient conditions are given for the polynomials \(\{Q_n(x)\}^\infty_{n=0}\) such that the polynomials \(\{P_n(x)+\mu Q_n(x)\}^\infty_{n=0}\) are eigenfunctions of a linear differential or difference operator of the form \(L+\mu A\) with eigenvalues \(\{\lambda_n+\mu\alpha_n\}^\infty_{n=0}\).