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

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}α(n)q^n=\dfrac{1}{R^5\left(q\right)}=\left(\sum_{n=0}^{\infty}α^{\prime}(n)q^n\right)^{-1},\\ \sum_{n=0}^{\infty}β(n)q^n=\dfrac{R(q)}{R\left(q^{16}\right)}=\left(\sum_{n=0}^{\infty}β^{\prime}(n)q^n\right)^{-1}, \end{align*} then \begin{align*} α(5n+r)&=-α^{\prime}(5n+r-2) \quad r\in\{3,4\},\\ β(10n+r)&=-β^{\prime}(10n+r-6) \quad r\in\{7,9\}. \end{align*}