Geodesic
Infinite-order differential equations with analytic solutions
Matematičeskie zametki, Tome 10 (1971) no. 3, pp. 269-278 
A. F. Leont'ev. Infinite-order differential equations with analytic solutions. Matematičeskie zametki, Tome 10 (1971) no. 3, pp. 269-278. http://geodesic.mathdoc.fr/item/MZM_1971_10_3_a3/
@article{MZM_1971_10_3_a3,
     author = {A. F. Leont'ev},
     title = {Infinite-order differential equations with analytic solutions},
     journal = {Matemati\v{c}eskie zametki},
     pages = {269--278},
     year = {1971},
     volume = {10},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1971_10_3_a3/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

An infinite-order linear differential equation with constant coefficients and characteristic equation of the class $[1, 0]$ is investigated, and a class of solutions is introduced. It is shown that, if the zeros $\lambda_k=\alpha_k+i\beta_k$ of the characteristic function satisfy the condition $\lim\limits_{k\to\infty}\left|\frac{\lambda_k}{\beta_k}\right|>0$, then all solutions of the class under consideration are analytic functions.