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On the Zeros of Certain Entire Functions

Ruiming Zhang

Abstract

In this work we prove that certain entire $q$-functions have infinitely many nonzero roots $\left\{ ρ_{n}\right\} _{n=1}^{\infty}$, as $n\to+\infty$ the moduli $\left|ρ_{n}\right|$ 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 $\mathcal{E}_{q}(z,t)$.

On the Zeros of Certain Entire Functions

Abstract

In this work we prove that certain entire -functions have infinitely many nonzero roots , as the moduli grow at least exponentially. Applications to entire -functions defined by series expansions are provided. These functions include the -analogue of the plane wave function .
Paper Structure (4 sections, 4 theorems, 62 equations)

This paper contains 4 sections, 4 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Applications
  4. Application to $\mathcal{E}_{q}(z;t)$

Key Result

Lemma 1

If such that then for any $r>0$, where $\theta(q)=\sum_{n\in\mathbb{Z}}q^{n^{2}/2}$. Consequently,

Theorems & Definitions (9)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Corollary 3
  • proof
  • Example 4
  • Corollary 5
  • proof