Normal solvability of a class of differential equations of infinite order

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 $0<Q\leqslant+\infty$ and $Q$ is determined by the coefficients $a_s^k$.
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