Gerard-Sibuya's versus Majima's concept of asymptotic expansion in severalvariables

We give an example of a holomorphic function. admitting Gerard-Sibuya asymp totic expansion on a polysector of C-n, and such that none of its derivativ es admits such an expansion. This motivates the study of the relationship b etween the concepts of asymptotic expansion in several variables respective ly given by Gerard-Sibuya and Majima. For a function f, Majima's notion is proved to be equivalent, on the one hand, to the existence of Gerard-Sibuya asymptotic expansion for f and its derivatives, and, on the other hand, to the boundedness of the derivatives of f on bounded proper subpolysectors o f S.