Results in Mathematics

, 74:52 | Cite as

On Zeros of Some Entire Functions



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


zeros of entire functions Pólya frequence sequence Vitali’s theorem Hurwitz’s theorem Rogers–Szegő polynomials Stieltjes–Wigert polynomials 

Mathematics Subject Classification

30C15 33D15 30C10 



This work was partially supported by the National Natural Science Foundation of China (Grant No. 11801451).


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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCentral South UniversityChangshaPeople’s Republic of China

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