On identities of the Rogers-Ramanujan type (Q5902991)

The author proves two transformation formulae for q-series: \[ \sum a_ k\left[ \begin{matrix} a+b\\ a+k\end{matrix} \right] \left[ \begin{matrix} a+b\\ b+k\end{matrix} \right]=\sum \frac{(q)_{a+b} q^{j^ 2} S_{2j}}{(q)_{a-j} (q)_{b-j} (q)_{2j}} \] where \(S_{2j}=\sum \left[ \begin{matrix} 2j\\ j+k\end{matrix} \right] q^{-k^ 2} a_ k\), and \[ \sum b_ k\left[ \begin{matrix} a+b+1\\ a+k\end{matrix} \right] \left[ \begin{matrix} a+b+1\\ b+k\end{matrix} \right]=\sum \frac{(q)_{a+b+1} q^{j^ 2+j} T_{2j+1}}{(q)_{a-j} (q)_{b-j} (q)_{2j+1}} \] where \[ T_{2j+1}=\sum \left[ \begin{matrix} 2j+1\\ j+k\end{matrix} \right] q^{-k^ 2+k} b_ k\quad. \] These are then used to derive many q-series identities including results by Rogers, Ramanujan, Slater, Göllnitz, Gordon, Andrews and this reviewer as well as some new results for moduli of the form \(4k+1\).