Solution of Inhomogeneous Differential Equations with Polynomial Coefficients in Terms of the Green's Function and the AC-Laplace Transform

The particular solutions of inhomogeneous differential equations with polynomial coefficients in terms of the Green's function are obtained in the framework of the AC-Laplace transform, that is the Laplace transform supplemented by its analytic continuation. In particular, discussions are given on Kummer's and the hypergeometric differential equation, and also on a fractional differential equation with coefficients of polynomial of at most first degree.