On Zeros of Some Entire Functions

Abstract

Let

$$\begin{aligned} A_{q}^{(\alpha )}(a;z)=\sum _{k=0}^{\infty }\frac{(a;q)_{k}q^{\alpha k^2} z^k}{(q;q)_{k}}, \end{aligned}$$

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.