Linear perturbations of differential or difference operators with polynomials as eigenfunctions

[摘要] This paper deals with one-parameter linear perturbations of a family of polynomials {P-n(x)}(infinity)(n=0) with deg[P-n(x)] = n of the form P-n(mu)(x) = P-n(x) + mu Q(n)(x), where mu is a real parameter and {Q(n)(x)}(infinity)(n=0) are polynomials with deg[Q(n)(x)] less than or equal to n. Let the polynomials {P-n(x)}(infinity)(n=0) be eigenfunctions of a linear differential or difference operator L with eigenvalues {lambda(n)}(infinity)(n=0). The purpose of this paper is to derive necessary and sufficient conditions for the polynomials {Q(n)(x)}(infinity)(n=0) such that the polynomials {P-n(mu)(x)}(infinity)(n=0) are eigenfunctions of a linear difference or differential operator (possibly of infinite order) of the form L + mu A with eigenvalues {lambda(n) + mu alpha(n)}(infinity)(n=0).

[发布日期] 1997-02-03

[关键词] differential operators;difference operators;orthogonal polynomials