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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.

30B99 Series expansions of functions of one complex variable
39A11 Stability of difference equations (MSC2000)
40G10 Abel, Borel and power series methods