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I am grateful to Y. Sibuya and B.L.J. Braaksma, who introduced me to the beautiful theory of multisummability in a joint seminar at the University of Minneapolis, in Spring of 1990, and I pray that they will consider me a good student. I also owe thanks to my student Andreas Beck, who went over the proofs and did all exercises, and to Sabine Lebhart for carefully typing the manuscript. Finally, I wish to apologize to my wife Christel, for spending parts of our vacation on a Dutch island on writing this text.

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Abstract. In an earlier paper, the first author showed that certain normalized formal solutions of homogeneous linear partial di¤erential equations with constant coe‰cients are multisummable, with a multisummability type that can be determined from a Newton polygon associated with the PDE. In this article, some of the results obtained there are extended in several directions: First of all, arbitrary formal solutions of inhomogeous PDE are considered, and it is shown that, in some sense, they can be computed completely explicitly. Secondly, the Gevrey order of these formal solutions is determined. Finally, formal power series are discussed that, in general, do not satisfy a PDE with constant coe‰cients, but instead may be considered as solutions of singularly perturbed ODE, or integro-di¤erential equations of a certain form.