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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
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