Multisummable Power Series

Abstract

In previous chapters we have shown that HLFFS are very natural to consider when discussing Stokes’ phenomenon, since they are κ-summable, for some suitable κ > 0. Unfortunately, their computation is not so easy, because of one step requiring use of Banach’s fixed point theorem. So in applications one may prefer to work with formal fundamental solutions in the classical sense. They can be computed relatively easily, using computer algebra tools which will briefly be discussed in Section 13.5. In Section 8.4 we have shown these FFS to be a product of finitely many matrix power series, each of which is κ-summable with a value of k depending on the factor. However, this factorization is neither unique nor fully constructive, so the problem of summation of FFS remains. In this chapter we are now presenting a summability method that is stronger than κ- summability for every κ > 0, enabling us to sum FFS as a whole. This method, named multisummability, was first introduced in somewhat different form by Ecalle [94], using what he called acceleration operators. Here, we present an equivalent definition, based on the more general integral operators introduced in Sections 5.5 and 5.6. We also show that it would be sufficient to work with Laplace operators only, but it can be convenient in applications to have the more general integral operators at hand: Sometimes one will meet formal power series for which it will be simpler to show applicability of some particular integral operator, but more complicated to do so for a Laplace operator, although theoretically they are equivalent.