Tea or Coffee: Please bring your own.
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.
The next speaker in our seminar is Shishuo Fu of Chongqing University, PRC. It may be Fool's day, but we're not kidding. It really is Shishuo who has consented to give a talk all the way from China!
The live broadcast did not work as anticipated in the previous talk; I hope it works this time. At any rate, its best to try and come for the zoom session.
Talk Announcement
Title: Bijective recurrences for Schroeder triangles and Comtet statistics
