On zeros of some entire functions

TL;DR

This paper investigates the zeros of a class of entire functions, including a specific q-series, providing partial answers to when these functions have only real zeros, and extends results to functions involving Rogers-Szeg ext{"o} and Stieltjes-Wigert polynomials.

Contribution

It establishes new conditions under which certain entire functions, including q-series and polynomial-related functions, have only real or infinitely many negative zeros.

Findings

The function $A_{q}^{(eta)}(q^l;z)$ has only infinitely many negative zeros for $l \\geq 2$.

Provides partial answers to Zhang's question on the reality of zeros of $A_{q}^{(eta)}(a;z)$.

Extends zero distribution results to functions involving Rogers-Szeg ext{"o} and Stieltjes-Wigert polynomials.

Abstract

Let \begin{equation*} A_{q}^{(\alpha)}(a;z) = \sum_{k=0}^{\infty} \frac{(a;q)_{k} q^{\alpha k^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 $A_{q}^{(\alpha)}(q^l;z),~l\geq 2$ 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-Szeg\H{o} polynomials and the Stieltjes-Wigert polynomials.