The classical results, concerned with the integration operator, are considered from the Gevrey formal power series point of view. Using the notion of the Cauchy-Laurent integral, the integral representation of the Dickson type is obtained for some differential operators of infinite order.

[1] Korobeinik Y.F. Shift operators on numerical families. - Rostov: Iz-vost Rostov Un-ta, 1983. - 154 p.

[2] Fage M.K., Nagnibida N.I. The problem of equivalence of ordinary linear differential operators. - Novosibirsk : Nauka, 1987. - 150 p.

[3] Gorodetsky V.V. Cauchy problems for evolutionary equations of infinite order. - Chernivtsi: Ruta, 2005.

[4] Yasutaka Sibuya. Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation. - Providence: AMS, 1990.

[5] D.Sauzin. Mould expansions for the saddlenode and resurgence  monomials. Renormalization and Galoistheories // IRMALect. Math. Theor. Phys.,15 ,Eur. Math. Soc., Zürich, 2009, р.83-163.

[6] S.Grabiner. A formal power series operational calculus for quasinilpotent operators // Duke Math. J.,no.38 (1971),p.641-658.

[7] Hefter S.L., Mokrenyuk V. Power series $\sum _{n=0}^∞ n!z^n$ and holomorphic solutions of some differential equations in Banach space // Journal of Mathematical Physics, Analysis and Geometry , 1 ( 2005 ) , p . 53 - 70.

[8] Linchuk S.S. Current applicability of integral operators of infinite order to some classes of integer functions // Scientific Bulletin of Cenovetsky University, issue 349, Mathematics, 2007, p. 70-73.

[9] D.G. Dickson. Infinite order differential equations // Proceedings of the Amer.Math.Soc.,Vol.15,No.4 (Aug.,1964), р.638-641.

[10] Helemsky A.J. Banach and polynormalised algebras : a general theory of homology representation . - Moscow : Nauka , 1989. - 464 p .

[11] Burbaki N. Topological vector spaces .- Moscow : Izd-vo foreign litt. 1959. 410 p.