Linear perturbations of differential of difference operators with polynomials as eigenfunctions

This paper deals with one-parameter linear perturbations of a family of polynomials {Pn(x)}n=0∞ with deg[Pn(x)] = n of the form Pnμ(x) = Pn(x) + μQn(x), where μ is a real parameter and {Qn(x)}n=0∞ are polynomials with deg[Qn(x)] ⩽ n. Let the polynomials {Pn(x)}n=0∞ be eigenfunctions of a linear differential or difference operator L+μA with eigenvalues {λn}n=0∞. The purpose of this p to derive necessary and sufficient conditions for the polynomials {Qn(x)}n=0∞ such that the polynomials {Pnμ(x)}n=0∞ are eigenfunctions of a linear difference or differential operator (possibly of infinite order) of the form L + μA with eigenvalues {λn + μαn}n=0∞.