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

TL;DR

This paper investigates the matching coefficients in the series expansions of specific $q$-products and their reciprocals, revealing new relations connected to Ramanujan's continued fractions and their properties.

Contribution

It introduces novel identities linking coefficients of $q$-product series and their reciprocals, especially in the context of Ramanujan's continued fractions.

Findings

Identifies matching coefficient relations for certain $q$-products and their reciprocals.

Establishes connections between these coefficients and Ramanujan's continued fractions.

Provides explicit formulas for coefficients in terms of modular parameters.

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*}