Resurgent functions and nonlinear systems of differential and difference equations (Q2161305)

The resurgent analysis, developed in [\textit{J. Ecalle}, Publ. Math. Orsay 81--05, 247 p. (1981; Zbl 0499.30034); Publ. Math. Orsay 81--06, 248--531 (1981; Zbl 0499.30035); Publ. Math. Orsay 85--05, 585 p. (1985; Zbl 0602.30029)] is an important tool to provide effective methods for the study of holomorphic dynamics analytic differential equations, WKB analysis, etc. Its key notion is the \textit{convolution product of endlessly continuable functions} which allows to describe their sheet structure in order to use the alien calculus. In practical cases, these functions are defined by the Borel transforms of convenient formal power series. In the present paper, the formal power series considered are the solutions of some nonlinear systems of differential and difference equations. After discussing endless continuability of iterated convolution products of resurgent functions associated with rooted trees, the author proves in particular that these formal solutions are all resurgent.