Introduction à la multisommabilité. (Introduction to multisummability). (French) Zbl 0722.34005

The paper is an informal introduction into the “multisummability theory” [J. Martinet and J.-P. Ramis, Computer algebra and differential equations, Colloq., Comput. Math. Appl., 117-214 (1988; Zbl 0722.12007)] and its applications in the theory of linear differential equations in the complex domain. From the fundamental formal matrix solution \(\hat Y(t)=\hat H(t)t^ L \exp Q(1/t)\) of a linear differential system \(Y'=A(x)Y\) where \(t=x^{1/p}\) is a suitable ramification, one may obtain a “true solution” by summation of the formal matrix \(\hat H.\) It is suggested that there exists a factorization of the matrix \(\hat H=\hat H_ 1\hat H_ 2...\hat H_ r\) such that \(\hat H_ i\) is \(k_ i\)- summable for every \(i=1,2,...,r\). The theory is illustrated by several simple examples that are analyzed in some detail.


34M99 Ordinary differential equations in the complex domain
40G10 Abel, Borel and power series methods
12H05 Differential algebra


Zbl 0722.12007