Convergence of a series of exponentials associated with an entire function
Let L(t) be an entire function of order \(\nu >1\), \(L(t)=\sum^{\infty}_{0}c_ kt^ k\) with simple zeros \(\{\lambda_ m\}\) such that \[ \underline{\lim}_{m\to \infty}\frac{\ln^+ \ln^+| L'(\lambda_ m)|}{\ln | \lambda_ m|}=\frac{\sigma}{\sigma -1},\quad \sigma \geq \frac{\nu}{\nu -1}. \] For any entire function F(z) of order \(<\nu /(\nu -1)\) let \[ w(z,L,F)=\sum c_ k(F^{(k-1)}(0)+zF^{(k-2)}(0)+...+z^{k- 1}F(0)),\quad a_ m=w(\lambda_ m,L,F)/L'(\lambda_ m). \] Associate with any such F a series \(\sum^{\infty}_{1}a_ m\exp (\lambda_ mz)\). Denote \(F_ n(z)\) the nth partial sum of this series and let \(\rho ((F_ n))=\inf \{\mu:\) there exists r(\(\mu)\), for all z, \(| z| >r(\mu)\), for all n, \(| F_ n(z)| <e^{| z|^{\mu}}\}.\) It is known that \(\rho ((F_ n))\in [1,\sigma]\). In the paper the following question is considered: is it possible to find for any \(\alpha\in [1,\sigma]\) a function F(z) of order \(<\nu /\nu -1\) such that \(\rho ((F_ n))=\alpha\). It is shown that there are no functions F(z) with \(\rho ((F_ n))=\alpha\) when \(\nu /\nu -1\leq \alpha <\sigma\).
scientific article; zbMATH DE number 3962063