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A particular type of summability of divergent power series, with an application to difference equations. (English) Zbl 0981.30002
The author studies difference equations like $$y(z+1) - a(z)y(z) = b(z)$$ with the coefficients analytic in a neighbourhood of $$\infty$$. The author aims at the case when the corresponding homogeneous equation has a solution being an entire function of order one and maximum type like $$y(z+1) - a(z)y(z) = 0$$ having a solution $$\frac{1}{\Gamma (z)}$$. If one looks for a formal solution of a non-homogeneous equation as a series in $$\frac{1}{z}$$, this series may have zero radius of convergence. In the article under review the author studies a method of summability of such formal series (”a weak Borel-sum”) and consider its applications to the difference equations as above.

##### MSC:
 30B99 Series expansions of functions of one complex variable 39A11 Stability of difference equations (MSC2000) 40G10 Abel, Borel and power series methods
##### Keywords:
difference equations; formal power series; Borel summation