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 $\mathcal{E}_{q}(z,t)$.