On the Zeros of Certain Entire Functions

Abstract

In this work we prove that certain entire $q$-functions have infinitely many nonzero roots ${\left\{{\rho }_n\right\}}_{n=1}^{\infty }$, as $n\to +\infty$ the moduli $\left|{\rho }_n\right|$ grow at least exponentially. Applications to entire $q$-functions defined by series expansions are provided. These functions include the $q$-analogue of the plane wave function ${\mathcal{E}}_q(z,t)$.

10 pages