On analytic continuation of power series with the condition on their Gelfond-Leontev derivatives

Abstract: For $R\in[0,\infty]$ we denote by $A(R)$ the class of power series $f(z)=\sum_{k=0}^{\infty} f_kz^k$, having radius of convergence $R$, and say that $f\in A^+(R)$ if $f\in A(R)$ with $f_k>0$. For $f\in A(0)$ and $l\in A^+(0)$ we put $D_l^nf(z)=\sum_{k=0}^{\infty} \frac {l_k}{l_{k+n}}f_{k+n}z^k$. The operator $D_l^nf$ is called the Gelfond--Leontev operator. If $l(z)=e^z$, then $D_l^n=\frac {d^n}{dz^n}$ is usual differential operator.

Let ${\Cal N}$ be a class of increasing sequences $\{n_p\}$ of positive integers, $n_0=0$, let $\Lambda$ be the class of all positive sequences $\lambda=(\lambda_k)$ and let $\Lambda^*=\{\lambda\in\Lambda: \ln \lambda_k=O(k)\}$ We say that $f\in A_{\lambda}(R)$ if $f \in A(R)$ and $|f_k|\le \lambda_k|f_1|$ for all $k \in {\Bbb N}$. Typical is

\proclaim{Theorem 1} Suppose that $l\in A^+(\infty)$ and $l_k^2/(l_{k-1}l_{k+1}) \searrow c \ge 0$. In order that for every $(n_p)\in{\Cal N}, \lambda \in \Lambda^*$ and $f\in A(0)$ the inclusion $D_l^{n_p}f\in A_{\lambda}(0)$ for all $p\in {\Bbb Z}_+$ implies the inclusion $f\in A(\infty)$ it is necessary and sufficient that $$ \frac 1{k+1}\ln\frac 1{l_{k+1}} - \frac 1k \ln \frac 1{l_k} \to \omega > 0, \quad k\to \infty. $$ \endproclaim

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