On the solution of some distributional differential equations: existence and characterizations of the classical moment functionals

Abstract

Given two polynomials φ and ψ such that deg φ ≤ 2 and deg ψ = 1, and a continuous linear functional U (defined on the space of all polynomials with complex coefficients) which is a solution of the differential distributional equation D(φU) = ψu, we determine necessary and sufficient conditions, involving only the polynomials φ and ψ, for the regularity of U, that is, for the existence of an orthogonal polynomial system associated with U. This result gives a new definition (or characterization) of classical orthogonal polynomials. We state some consequences. Moreover, some interesting relations related to the parameters which appears in some known characterizations of the classical orthogonal polynomials are obtained in an unified way.