Geodesic
Differential equations of infinite order and infinite systems of
Izvestiya. Mathematics , Tome 4 (1970) no. 4, pp. 891-930.

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In this article a one-to-one correspondence is established between solutions of differential equations of infinite order with constant coefficients and infinite systems of first-order linear differential equations. With the aid of this correspondence new results in the theory of analytic solutions of differential equations of infinite order are obtained. 
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Yu. F. Korobeinik. Differential equations of infinite order and infinite systems of. Izvestiya. Mathematics , Tome 4 (1970) no. 4, pp. 891-930. http://geodesic.mathdoc.fr/item/IM2_1970_4_4_a9/

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