Further Applications of Cubic $q$-Binomial Transformations
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| Format: | Preprint |
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2025
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| _version_ | 1866918440161247232 |
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| author | Berkovich, Alexander Dhar, Aritram |
| author_facet | Berkovich, Alexander Dhar, Aritram |
| contents | Consider \begin{align*} G(N,M;α,β,K,q) = \sum\limits_{j\in\mathbb{Z}}(-1)^jq^{\frac{1}{2}Kj((α+β)j+α-β)}\left[\begin{matrix}M+N\\N-Kj\end{matrix}\right]_{q}. \end{align*} In this paper, we prove the non-negativity of coefficients of some cases of $G(N,M;α,β,K,q)$. For instance, for non-negative integers $n$ and $t$, we prove that\\ \begin{align*} G\left(n,n;\frac{4}{3}+\frac{3(3^t-1)}{2},\frac{5}{3}+\frac{3(3^t-1)}{2},3^{t+1},q\right) \end{align*} and \begin{align*} G\left(n-\frac{3^t-1}{2},n+\frac{3^t+1}{2};\frac{8}{3}+2(3^t-1),\frac{4}{3}-(3^t-1),3^{t+1},q\right)\\ \end{align*} are polynomials in $q$ with non-negative coefficients. Using cubic positivity preserving transformations of Berkovich and Warnaar and some known formulae arising from Rogers-Szegö polynomials, we establish new identities such as\\ \begin{align*} \sum\limits_{0\le 3j\le n}\dfrac{(q^3;q^3)_{n-j-1}(1-q^{2n})q^{3j^2}}{(q;q)_{n-3j}(q^6;q^6)_{j}} = \sum\limits_{j=-\infty}^{\infty}(-1)^jq^{6j^2}{2n\brack n-3j}_q. \end{align*} |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_00130 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Further Applications of Cubic $q$-Binomial Transformations Berkovich, Alexander Dhar, Aritram Number Theory Combinatorics 05A15, 05A17, 05A30, 11P81, 11P84 Consider \begin{align*} G(N,M;α,β,K,q) = \sum\limits_{j\in\mathbb{Z}}(-1)^jq^{\frac{1}{2}Kj((α+β)j+α-β)}\left[\begin{matrix}M+N\\N-Kj\end{matrix}\right]_{q}. \end{align*} In this paper, we prove the non-negativity of coefficients of some cases of $G(N,M;α,β,K,q)$. For instance, for non-negative integers $n$ and $t$, we prove that\\ \begin{align*} G\left(n,n;\frac{4}{3}+\frac{3(3^t-1)}{2},\frac{5}{3}+\frac{3(3^t-1)}{2},3^{t+1},q\right) \end{align*} and \begin{align*} G\left(n-\frac{3^t-1}{2},n+\frac{3^t+1}{2};\frac{8}{3}+2(3^t-1),\frac{4}{3}-(3^t-1),3^{t+1},q\right)\\ \end{align*} are polynomials in $q$ with non-negative coefficients. Using cubic positivity preserving transformations of Berkovich and Warnaar and some known formulae arising from Rogers-Szegö polynomials, we establish new identities such as\\ \begin{align*} \sum\limits_{0\le 3j\le n}\dfrac{(q^3;q^3)_{n-j-1}(1-q^{2n})q^{3j^2}}{(q;q)_{n-3j}(q^6;q^6)_{j}} = \sum\limits_{j=-\infty}^{\infty}(-1)^jq^{6j^2}{2n\brack n-3j}_q. \end{align*} |
| title | Further Applications of Cubic $q$-Binomial Transformations |
| topic | Number Theory Combinatorics 05A15, 05A17, 05A30, 11P81, 11P84 |
| url | https://arxiv.org/abs/2510.00130 |