On the Zeros of Certain Entire Functions

In this work we prove that certain entire $q$-functions have infinitely many nonzero roots ${\left\{{\rho }_n\right\}}_{n=1}^{\mathrm{\infty }}$, as $n\to +\mathrm{\infty }$ the moduli $\mid {\rho }_n\mid$ 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 ${\mathcal{E}}_q(z,t)$.