A skew polynomial approach to integro-differential operators

Abstract

We construct the algebra of integro-differential operators over an ordinary integro-differential algebra directly in terms of normal forms. In the case of polynomial coefficients, we use skew polynomials for defining the integro-differential Weyl algebra as a natural extension of the classical Weyl algebra in one variable. Its normal forms, algebraic properties and its relation to the localization of differential operators are studied. Fixing the integration constant, we regain the integro-differential operators with polynomial coefficients.