Six Equivalent Approaches to Multisummability

Abstract

We consider the case of divergent series that are, in some way, relevant to several levels of summation together. We introduce the problems inherent to the situation with the example of the Ramis-Sibuya series and we show that this series is k-summable for no k > 0. We expound six different theories of multisummability which extend the theories of k-summability. In most of them we found useful to treat the case when the summability depends on only two levels k ₁ < k ₂ before to state general results depending on an arbitrary number of levels. We prove the equivalence of the relativeWatson’s lemma with the Tauberian theorem proved in Chapter 5. As an application we prove that any solution of a linear ordinary differential equation is multisummable for convenient levels $k_1 < k_2 < \cdots < k_v$ of summation.