Further Applications of Cubic $q$-Binomial Transformations
Alexander Berkovich, Aritram Dhar
TL;DR
The paper addresses nonnegativity of coefficients in the two-parameter $q$-series $G(N,M;\alpha,\beta,K,q)$ and the companion $D_{K,i}(N,M;\alpha,\beta)$, tying these questions to Borwein's mod 3 conjectures. It develops and applies cubic positivity-preserving transformations from Berkovich-Warnaar together with Rogers-Szego polynomials to produce new nonnegative $q$-polynomials and to derive Rogers-Szego based identities that relate alternating $q$-series to sums with nonnegative coefficients. It proves nonnegativity for specific parameter families, such as $G\left(n,n;\frac{4}{3}+\frac{3(3^t-1)}{2},\frac{5}{3}+\frac{3(3^t-1)}{2},3^{t+1},q\right)$ and related forms, and presents identities like a cubic-transform equality to a nonnegative right-hand side involving a $q$-binomial sum. These results enhance positivity techniques in $q$-series and advance understanding of mod 3 phenomena in Borwein-type conjectures.
Abstract
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*}