The Bailey-Rogers-Ramanujan group (Q2782405)

A pair of sequences \((\alpha_n(a,q),\beta_n(a,q))\) is called a Bailey pair with parameters \((a,q)\) if NEWLINE\[NEWLINE\beta_n(a,q) = \sum_{r=0}^n \frac{\alpha_n(a,q)}{(q;q)_{n-r}(aq;q)_{n+r}}NEWLINE\]NEWLINE for all \(n\geq 0\). It is known that Bailey's Lemma takes a Bailey pair NEWLINE\[NEWLINE(\alpha_n(a,q),\beta_n(a,q))NEWLINE\]NEWLINE and produces another Bailey pair NEWLINE\[NEWLINE(\alpha_n'(a,q),\beta_n'(a,q)).NEWLINE\]NEWLINE Various limiting cases of the Bailey's Lemma allow the author to establish a link between transformations of Bailey pairs via the Bailey's Lemma and the elements in the group (the author called this group the Bailey-Rogers-Ramanujan group) generated by NEWLINE\[NEWLINES_1=\left(\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right), \quad S_2=\left(\begin{matrix} 1 & 1/2 \\ 0 & 1\end{matrix}\right), \quad D_1=\left(\begin{matrix} 1 & 0 \\ 0 & 2\end{matrix}\right), \quad D_2=\left(\begin{matrix} 1 & -1 \\ 0 & 2\end{matrix}\right),NEWLINE\]NEWLINE NEWLINE\[NEWLINED_3=\left(\begin{matrix} 1 & -1/2 \\ 0 & 2\end{matrix}\right), \quad E_1=\left(\begin{matrix} 1 & 0 \\ 0 & 1/2\end{matrix}\right), \quad E_2-\left(\begin{matrix} 1 & 1/2 \\ 0 & 1/2\end{matrix}\right),NEWLINE\]NEWLINE NEWLINE\[NEWLINEE_3=\left(\begin{matrix} 1 & 1/4 \\ 0 & 1/2\end{matrix}\right), \quad T_1= \left(\begin{matrix} 1 & 1/3 \\ 0 & 1/3\end{matrix}\right)\quad\text{and}\quad T_2=\left(\begin{matrix} 1 & -1 \\ 0 & 3\end{matrix}\right).NEWLINE\]NEWLINE More precisely, for any \(\omega = \omega_1\omega_2\cdots \omega_{k+1}\) in the Bailey-Rogers-Ramanujan group, there is a finite Rogers-Ramanujan identity corresponding to \(\omega\). Examples due to L. Slater, as well as new Rogers-Ramanujan type identities are given in Section 5. At the end of the article, alternative representations of the Borwein polynomials are given. NEWLINENEWLINENEWLINEThis is a very well written paper and I highly recommend it to anyone who are interested in Rogers-Ramanujan type identities.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00024].