Geodesic
Normal solvability of a class of differential equations of infinite order
Sbornik. Mathematics, Tome 13 (1971) no. 3, pp. 371-399 
Yu. F. Korobeinik; O. V. Epifanov. Normal solvability of a class of differential equations of infinite order. Sbornik. Mathematics, Tome 13 (1971) no. 3, pp. 371-399. http://geodesic.mathdoc.fr/item/SM_1971_13_3_a3/
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this article we study a differential equation of infinite order with polynomial coefficients \begin{equation} Ly\equiv\sum^\infty_{k=1}P_k(x)y^k(x)=f(x),\qquad P_k(x)=\sum^{n_k}_{s= 0}a_s^k x^s, \end{equation} where $\varlimsup_{k\to\infty}\frac{n_k}k=\alpha<1$. Under given conditions on the coefficients $a_s^k$, normal solvability of equation $(1)$ is established in the class of entire functions $[1-\alpha,Q]$, where $0 and $Q$ is determined by the coefficients $a_s^k$. Bibliography: 10 titles.

[1] Yu. F. Korobeinik, “Nekotorye primeneniya teorii normalno-razreshimykh operatorov k differentsialnym uravneniyam beskonechnogo poryadka”, Matem. sb., 72(114) (1967), 3–37 | MR | Zbl

[2] Yu. F. Korobeinik, “Normalno-razreshimye operatory i differentsialnye uravneniya beskonechnogo poryadka”, Litov. matem. sb., XI:3 (1971)

[3] O. Perron, “Lineare Differentialgleichungen unendlich hoher Ordnung mit ganzen rationalen Koeffitienten”, Math. Ann., 84 (1921), 31–41 | DOI | MR

[4] A. F. Leontev, “Differentsialno-raznostnye uravneniya s lineinymi koeffitsientami”, Trudy ped. in-ta, 14, Gorkii, 1951, 3–30

[5] M. G. Krein, I. Ts. Gokhberg, “Osnovnye polozheniya o defektnykh chislakh, kornevykh chislakh i indeksakh lineinykh operatorov”, Uspekhi matem. nauk, XII:2(74) (1957), 43–118 | MR

[6] N. Burbaki, Obschaya topologiya. Osnovnye struktury, Fizmatgiz, Moskva, 1958

[7] Yu. F. Korobeinik, “O primenimosti differentsialnykh operatorov beskonechnogo poryadka”, Sib. matem. zh., X:3 (1969), 549–564 | MR

[8] Yu. F. Korobeinik, “Ob odnom klasse uravnenii beskonechnogo poryadka v obobschennykh proizvodnykh”, Litov. matem. sb., IV:4 (1964), 497–515 | MR

[9] Yu. F. Korobeinik, “O preobrazovaniyakh analiticheskikh prostranstv s pomoschyu differentsialnykh operatorov beskonechnogo poryadka”, Uspekhi matem. nauk, XX:5(125) (1965), 208–213 | MR

[10] P. van der Steen, On differential operators of infinite order, Doctoral dissertation, Technical University of Delft, Uitgeverij Waltman, Delft, 1968 | MR | Zbl