Power series methods and multisummability. (English) Zbl 1078.34534
Summary: We recall the notion of a power series summability method and show, using results of Balser and A. Beck, that existence of a power summability method which is stronger than multisummability would be equivalent to existence of an entire function with certain particular properties. We then prove that such entire functions cannot exist. In contrast to this negative result, we show that an iteration of power series methods is indeed stronger than multisummability.
MSC:
34M30 | Asymptotics and summation methods for ordinary differential equations in the complex domain |
30D15 | Special classes of entire functions of one complex variable and growth estimates |
34A25 | Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. |
40D25 | Inclusion and equivalence theorems in summability theory |