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On the positive coefficients of two families of $q$-series

Ji-Cai Liu, Kong-Lian Liao

Abstract

Let $S$ be a finite set of pairwise coprime positive integers and $Ax^2+Bx$ be an integer valued polynomial with $A> B\ge 0$. For integers $k\ge 1$ and $n\ge 0$, the coefficients $γ_{S,A,B}^k (n)$ are defined as \begin{align*} \prod_{s\in S}\frac{1}{1-q^s}\sum_{j\not\in [-k,k-1]} (-1)^{j+k}q^{Aj^2+Bj}=\sum_{n= 0}^{\infty}γ_{S,A,B}^k (n)q^n. \end{align*} In this paper, we investigate the positivity of $γ_{S,A,B}^k (n)$ for $|S|=4,5$.

On the positive coefficients of two families of $q$-series

Abstract

Let be a finite set of pairwise coprime positive integers and be an integer valued polynomial with . For integers and , the coefficients are defined as \begin{align*} \prod_{s\in S}\frac{1}{1-q^s}\sum_{j\not\in [-k,k-1]} (-1)^{j+k}q^{Aj^2+Bj}=\sum_{n= 0}^{\infty}γ_{S,A,B}^k (n)q^n. \end{align*} In this paper, we investigate the positivity of for .

Paper Structure

This paper contains 5 sections, 9 theorems, 66 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Proof of Theorems \ref{['t-1']}
  5. Proof of Theorem \ref{['t-2']}

Key Result

Theorem 2.1

Let $Y=\{y_1,y_2,y_3,y_4\}$ in which $y_1,y_2,y_3,y_4$ are pairwise coprime positive integers. Then $\gamma_{Y,A,B}^k (n)\ge 0$ for $n\ge N_{Y,A,B}^k=A\left(k+2L_{Y,A,B}^k\right)^2+B\left(k+2L_{Y,A,B}^k\right)$ with where $H_1,H_2,\cdots,H_8$ are listed in Appendix A.

Theorems & Definitions (9)

  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Corollary 2.5
  • Corollary 2.6
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3