Summability in a direction of formal power series in several variables

Article type: Research Article

Authors: Sanz, Javier

Affiliations: Departamento de Análisis Matemático, Universidad de Valladolid, 47005 Valladolid, Spain E‐mail: [email protected]

Abstract: In this paper a theory of summability in a direction, extending that of J.P. Ramis, is developed for formal power series of several variables. To this end, generalized Laplace and Borel transforms are studied, as well as their action on functions admitting Gevrey strongly asymptotic expansion as defined by H. Majima. The definition we give of summability in a direction turns out to be, in a sense, equivalent to an iterative classical summation procedure. As an application we provide a new proof of a well‐known result of R. Gérard and Y. Sibuya stating the convergence of the formal power series solution to a certain completely integrable Pfaffian system.

Journal: Asymptotic Analysis, vol. 29, no. 2, pp. 115-141, 2002