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In 1978-80 the first author introduced the notion of A;-summability of formal power series expansions. This notion is based on fundamental works of Borel, Watson, Nevanlinna. He proved that power series solutions of sufficiently generic linear meromorphic differential equations are ksummable for some k > 0 depending on the equation. He remarked also that unfortunately there exists some formal power series solutions of some linear meromorphic differential equations which are not fc-summable for any k > 0 (cf. J.-P. Ramis and Y. Sibuya [10] for an example). J.-P. Ramis proved also a factorization theorem of formal solutions (J.-P. Ramis [9], Y. Sibuya [12] : Theorem 4.2.3, p. 237) which implies that every formal power series / solution of a linear meromorphic differential equation can (non uniquely) be written as a sum of products of fc-summable power series, where the occuring fc's belong to a finite set depending on the equation. Then it is possible to get a multisum f of / in a given generic direction d. Later J.-P. Ramis proved that this multisum does not depend on the decomposition and that, if d is fixed, the map / i—> / is Galois. (W. Balser obtained also independently the factorization theorem as a byproduct of his treatment of his first level formal solutions.)

A new proof of multisummability of formal solutions of non linear meromorphic differential equations