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arXiv · 2110.15546

Matching coefficients in the series expansions of certain $q$-products and their reciprocals

Nayandeep Deka Baruah·Hirakjyoti Das
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Abstract

We show that the series expansions of certain $q$-products have \textit{matching coefficients} with their reciprocals. Several of the results are associated to Ramanujan's continued fractions. For example, let $R(q)$ denote the Rogers-Ramanujan continued fraction having the well-known $q$-product repesentation $$R(q)=\dfrac{(q;q^5)_\infty(q^4;q^5)_\infty}{(q^2;q^5)_\infty(q^3;q^5)_\infty}.$$ If \begin{align*} \sum_{n=0}^{\infty}\alpha(n)q^n=\dfrac{1}{R^5\left(q\right)}=\left(\sum_{n=0}^{\infty}\alpha^{\prime}(n)q^n\right)^{-1},\\ \sum_{n=0}^{\infty}\beta(n)q^n=\dfrac{R(q)}{R\left(q^{16}\right)}=\left(\sum_{n=0}^{\infty}\beta^{\prime}(n)q^n\right)^{-1}, \end{align*} then \begin{align*} \alpha(5n+r)&=-\alpha^{\prime}(5n+r-2) \quad r\in\{3,4\},\\ \beta(10n+r)&=-\beta^{\prime}(10n+r-6) \quad r\in\{7,9\}. \end{align*}

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Nayandeep Deka Baruah, Hirakjyoti Das. 2021-10-29. Matching coefficients in the series expansions of certain $q$-products and their reciprocals. https://doi.org/10.1007/s11139-021-00534-4

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