The spectral values of a boundary value problem and the zeros of Mittag-Leffler functions (Q1874938)

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 \((-\infty, -(2N)!/N!)\).