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A different characterization of multi-summable power series. (English) Zbl 0759.40005
Author’s abstract: “Given $$k_ 1>k_ 2>k_ 3>\dots>k_ p>0$$ and a real number $$d$$, it is shown that a formal power series $$\hat f$$ is $$(k_ 1,\dots,k_ p)$$-summable in direction $$d$$ if and only if $$\hat f$$ can be decomposed into a sum of formal series $$\hat f_ j$$ (which may be series in a root of $$z$$, unless we restrict to $$k_ p\geq{1\over 2}$$), so that each $$\hat f_ j$$ is $$k_ j$$-summable in direction $$d$$, for $$j=1,\dots,p$$”.

##### MSC:
 40C15 Function-theoretic methods (including power series methods and semicontinuous methods) for summability 34E05 Asymptotic expansions of solutions to ordinary differential equations 30E15 Asymptotic representations in the complex plane
##### Keywords:
multi-summable power series; asymptotic expansion