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The power series $\sum_{n=0}^\infty n!\,z^n$ and holomorphic solutions of some differential equations in a Banach space
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Let $A$ be a bounded operator on a Banach space. A question about the existence of holomorphic solutions of the equation $z^2Aw'+g(z)=w$ is studied. Moreover, general properties of power series of the form $\sum_{n=0}^\infty c_nA^nz^n$, $c_n\in\mathbb C$ are considered. 
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S. L. Gefter; V. N. Mokrenyuk. The power series $\sum_{n=0}^\infty n!\,z^n$ and holomorphic solutions of some differential equations in a Banach space. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 1 (2005) no. 1, pp. 53-70. http://geodesic.mathdoc.fr/item/JMAG_2005_1_1_a2/

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