The Poincaré series of the module of derivations of affine monomial curves

Let A be a graded k-algebra and M be a finitely generated A-module. The Poincaré series is the formal power series . We study the Poincaré series of Derkk[S], the module of derivations of a numerical semigroup ring k[S], and we relate it to the Poincaré series of k over k[S] and to the type of S. We then use this in order to determine the Poincaré series of Derkk[S] or, at least, its rationality, for some classes of examples. We finally give an example of a non-rational .