Representation of entire functions by series in terms of Mittag-Leffler functions (Q790981)

Notations: \(E_{\rho}(z)=\sum z^ n/\Gamma(\rho^{-1}\quad n+1),\quad \rho>\frac{1}{2},\) (Mittag-Leffler function), \[ \Phi(z)=\sum | a_ nE_{\rho}(\lambda_ nz)|,\quad | \lambda_ n| \nearrow \infty \quad(assumed\quad convergent\quad for\quad all\quad z\in {\mathbb{C}}), \] \[ H(\theta)=\lim \sup_{r\to \infty}r^{-\rho_ 1} \log \Phi(re^{i\theta}),\quad \rho_ 1>\rho \quad(H(\theta)<\infty \quad by\quad assumption). \] Each function H is defined through the sequences \(a_ n,\lambda_ n\). Denote the class of such H by \(B_{\rho_ 1}\). The main result is as follows: For each \(H\in B_{\rho_ 1}\) there exists a sequence \(\mu_ n\) such that every entire function f admitting an indicator function \[ h(\theta)=\lim \sup_{r\to \infty}r^{-\rho_ 1} \log | f(re^{i\theta})| \] such that \(h(\theta)<H(\theta)\) can be expanded as \(f(z)=\sum A_ nE_{\rho}(\mu_ nz),\) where \[ \lim \sup r^{-\rho_ 1} \log F(re^{i\theta})\leq H(\theta),\quad F(z)=\sum | A_ nE_{\rho}(\mu_ nz)|. \] A formula is given for the coefficients \(A_ n\). This generalizes the corresponding result for Dirichlet series [\textit{A. F. Leont'ev}, Tr. Mat. Inst. Steklova 157, 68- 89 (1981; Zbl 0474.30004)]. The present paper contains considerable detailed information about indicator functions, certain contour integrals, etc. that cannot be easily summarized in a brief review.