On some representing systems in spaces of analytic functions

Let $E_\rho(z)$ be the Mittag-Leffler function. This article investigates the connection between “representing” properties for systems $\mathscr E_{\rho,\Lambda}=\{E_{\rho}(\lambda_kz)\}^{\infty}_{k=1}$ and $\mathscr E^{(n)}_{\rho,\Lambda}=\{E_\rho(\lambda_kz),zE_\rho(\lambda_kz),\dots,z^nE_\rho(\lambda_kz)\}^{\infty}_{k=1}$, $n\geqslant1$, as well as for systems $\mathscr E^1_{\rho,\Lambda}=\{E_\rho(\lambda_{k,1}z)\}^\infty_{k=1}$, $\mathscr E^2_{\rho,\Lambda}=\{E_\rho(\lambda_{k,2}z)\}^\infty_{k=1}$, and $\mathscr E^3_{\rho,\Lambda}=\mathscr E^1_{\rho,\Lambda}\cup\mathscr E^2_{\rho,\Lambda}$ in spaces of analytic functions.
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