On zeros of some entire functions

Abstract

Let \begin{equation*} A_{q}^{(α)}(a;z) = \sum_{k=0}^{\infty} \frac{(a;q)_{k} q^{αk^2} z^k} {(q;q)_{k}}, \end{equation*} where $\alpha >0,0<q<1.$ In a paper of Ruiming Zhang, he asked under what conditions the zeros of the entire function $A_q^{(\alpha )}(a;z)$ are all real and established some results on the zeros of $A_q^{(\alpha )}(a;z)$ which present a partial answer to that question. In the present paper, we will set up some results on certain entire functions which includes that $A_q^{(\alpha )}(q^l;z),l\ge 2$ has only infinitely many negative zeros that gives a partial answer to Zhang's question. In addition, we establish some results on zeros of certain entire functions involving the Rogers-Szegő polynomials and the Stieltjes-Wigert polynomials.

arXiv comment16pages