**Talk Announcement:**

**Title:**Matching coefficients in the series expansions of certain $q$-products and their inverses.

**Speaker:**Nayandeep Deka Baruah (Tezpur University)

**When:**April 15, 2021 - 3:55 PM - 5:00 PM (IST)

**Where:**Zoom: Please write to sf-and-nt@gmail.com for the link.

**Tea or Coffee**: Please bring your own.

**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.

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