On the positive coefficients of two families of $q$-series
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915008550535168 |
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| author | Liu, Ji-Cai Liao, Kong-Lian |
| author_facet | Liu, Ji-Cai Liao, Kong-Lian |
| contents | 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 $|S|=4,5$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_04726 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the positive coefficients of two families of $q$-series Liu, Ji-Cai Liao, Kong-Lian Number Theory Combinatorics 11S05, 05A17, 05A20 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 $|S|=4,5$. |
| title | On the positive coefficients of two families of $q$-series |
| topic | Number Theory Combinatorics 11S05, 05A17, 05A20 |
| url | https://arxiv.org/abs/2411.04726 |