A particular type of summability of divergent power series, with an application to difference equations (Q2730803)

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.