Matching coefficients in the series expansions of certain $q$-products and their reciprocals (2110.15546v1)

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*}