Integration of a Dirac comb and the Bernoulli polynomials

For any positive integer n, we consider the ordinary differential equations of the formy(n)=1−Ш+Fy(n)=1−Ш+F where Ш denotes the Dirac comb distribution and F   is a piecewise-C∞C∞ periodic function with null average integral. We prove the existence and uniqueness of periodic solutions of maximal regularity. Above all, these solutions are given by means of finite explicit formulae involving a minimal number of Bernoulli polynomials. We generalize this approach to a larger class of differential equations for which the computation of periodic solutions is also sharp, finite and effective.