Asymptotic zero distribution of sections and tails of Mittag-Leffler functions
Consider the Mittag-Leffler function \[ E_{1/ \rho}(z) = \sum^\infty_{j=0} \bigl(z^j/ \Gamma(1+j/ \rho)\bigr)= \left(\sum^n_{j=0} + \sum^\infty_{j=n+1} \right) \bigl(z^j/ \Gamma(1+j/ \rho)\bigr)=S_n(z,E_{1/ \rho})+t_{n+1} (z,E_{1/ \rho}) \] and let \[ R_n=\Gamma(1+n/ \rho)/ \Gamma(1+(n-1)/ \rho)\;{\mathcal M}_n(\lambda, E_{1/ \rho})= \{z:I_n(R_nz; \lambda,E_{1/ \rho})=0\} \] and \({\mathcal M}(\lambda, E_{1/\rho})\) the set of all accumulation points of \(\bigcup^\infty_{n=1} {\mathcal M}_n(\lambda, E_{1/ \rho})\), where \[ I_n(R_nz; \lambda,E_{1/ \rho})=(1-\lambda)s_n (R_nz,E_{1/ \rho}) -\lambda t_{n+1} (R_nz,E_{1/ \rho}). \] Let \[ \begin{aligned} S_1(\rho) &=\{z=re^{i\varphi} : r\leq 1, \varphi\in\Phi_r\} \cup\{z=re^{i\varphi}: r=e^{-1/\rho}, \pi/2\rho < \varphi<2 \pi-\pi /2\rho\},\\ S_2(\rho)& =\{z=re^{i\varphi}: r\geq 1,\varphi\in \Phi_x\},\;S(\rho)=S_1 (\rho)\cup S_2(\rho) \end{aligned} \] and \[ \Phi_x= \{\varphi: |\varphi|\leq\pi/2 \rho,\;r^\rho \cos(\varphi\rho)-1-\rho\log r=0\}. \] The author proves that \({\mathcal M}_0(\lambda, E_{1/ \rho})={\mathcal M}(\lambda,E_{1/ \rho})\setminus\{z: \arg z=\pm\pi/2 \rho\}\) is equal to \(S_1(\rho)\), \(S_2(\rho)\) or \(S(\rho)\) according to \(\lambda=0\), 1 or \(\lambda\in \mathbb{C}\setminus \{0,1\}\) respectively. This is an extension of \textit{G. Szegö}'s result on \(e^z\) [Sitzungsber. Berliner Math. Gesellsch. 23, 50-64 (1924; JFM 50.0257.01)] and completes the analogous extension given by \textit{A. Edrei}, \textit{E. B. Saff} and \textit{R. S. Varga} [Lect. Notes in Math. 1002, 1-115 (1983; Zbl 0507.30001)]. The latter's results on zero-free regions and zero distribution of \(I_n(R_nz; \lambda,E_{1/ \rho})\) are also extended.
scientific article; zbMATH DE number 1803915
scientific article