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

Abstract: We show that the series expansions of certain $q$-products have \textit{matching coefficients} with their inverses. 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 $R(q)=\left(q,q^4;q^5\right)_{\infty}/\left(q^2,q^3;q^5\right)_{\infty}$. 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\}\\ \text{and}&\\ \beta(10n+r)&=-\beta^{\prime}(10n+r-6), \quad r\in\{7,9\}. \end{align*} This is a joint work with Hirakjyoti Das.

classical analysis and ODEscombinatoricsnumber theory

Audience: researchers in the topic

Special Functions and Number Theory seminar

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