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

On the method of differentiation and its application to asymptotics for the heat kernel on H-type groups