The spectral values of a boundary value problem and the zeros of Mittag-Leffler functions
type: Item
instance of: scholarly article
title: The spectral values of a boundary value problem and the zeros of Mittag-Leffler functions
zbMATH Open document ID: 1035.34096
DOI: 10.1023/A:1020206623615
author: A. Yu Popov
published in: Differential Equations
publication date: 2003-05-25T00:00:00Z
review text: Let \(N\in\aleph\). For the differential equation \(y^{(N)}=\lambda y +1\) with the boundary conditions \(y^{k}(0)=0, 0\leq k\leq N-1\), \(y(1)=0\), the author studies the spectral value problem. It is known [\textit{I. V. Tikhonov} and \textit{Yu. S. Eidel'man}, Differ. Equ. 38, 669--677 (2002); translation from Differ. Uravn., 2002, 38, 637--644 (2002; Zbl 1033.34017)] that the set \(\Lambda_N\) of all spectral values of the problem above is the zero set of the entire function \(\chi_{N}(z)=\sum_{k=0}^{\infty}z/(kN+N)!\) which belongs to the family of Mittag-Leffler functions \(E_{\varrho}(z, \mu)\). For \(N\geq 3\), the set \(\Lambda_{N}\) has no simple analytic description and the following theorem is the main result of the paper in the description of \(\Lambda_{N}\). Theorem 1: For every \(N\in\aleph, N\geq 3\), all zeros of the functions \(\chi_{N}(z)\equiv E_{1/N}(z, N + 1)\) are real and simple and lie on the half-line \((-\infty, -(2N)!/N!)\).
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 34B05
zbMATH DE Number: 1916084
zbMATH Keywords: Mittag-Leffler functions
reviewed by: Mykola Grygorenko
MaRDI profile type: Publication
full work available at URL: a:1020206623615
OpenAlex ID: W180527096
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description: scientific article; zbMATH DE number 1916084