  The aim of this note is twofold. The first one is to find conditions on the
asymptotic sequence which ensures differentiation of a general asymptotic
expansion with respect to it. Our method results from the classical one but
generalizes it. As an application, our second aim is to give sharp asymptotic
estimates at infinity of the heat kernel on H-type groups by the method of
differentiation provided we have the result of the isotropic Heisenberg groups.
