Sums of exponential functions having only real zeros
Let \(G(z)\) be an entire function of genus 0 or 1 and \(a_m\) and \(b_m\) be positive numbers \((m=1,2,\dots,n)\). Let \(\sigma\) be any of the \(2^n\) vectors \((\pm,\pm, \dots,\pm)\), \(|\sigma|\) representing the number of plus signs in the vector \(\sigma\) and \(\sigma\cdot (ia_1, ia_2,\dots,ia_n)\) be the ordinary dot product. Let \[ H_n(w)=\Sigma G\bigl(\sigma\cdot (ia_1,\dots,ia_n) \bigr)^{w\bigl(\sigma \cdot(ib_1, \dots,ib_n)\bigr)} \] and \[ P_n(t)=\Sigma G\bigl(\sigma\cdot (ia_1, \dots, ia_n)\bigr)t^{|\sigma|}, \] where the summation are over all \(\sigma\) and in each term in the summations the same \(\sigma\) is used both in the argument of \(G\) and in \(w(\cdot)\) or \(t^{|\cdot |}\). The author proves that all the zeros of \(H_n(w)\) are real and all the zeros of \(P_n(t)\) are on the unit circle in \(\mathbb{C}\).
scientific article; zbMATH DE number 2078014