New finite Rogers-Ramanujan identities
Let \(q\) be a complex number with \(0<|q|<1\). For any integer \(n\) define the \(q\)-shifted factorial \((a)_n\) by \((a)_0=1\) and \[ (a)_n=(a;q)_n:=\begin{cases}(1-a)(1-aq)\dots(1-aq^{n-1}),&n=1,2,\dots,\\ [(1-aq^{-1})(1-aq^{-2})\dots(1-aq^n)]^{-1},&n=-1,-2,\dots.\end{cases} \] For \(m\geq 1\) define: \[ (a_1,\dots,a_m)_n:=(a_1)_n\dots (a_m)_n,\quad (a_1,\dots,a_m)_\infty:=\lim_{n\to\infty}(a_1,\dots,a_m)_n. \] The authors prove: \textbf{Theorem.} If at least one of \(a\), \(b\) and \(c\) is of the form \(q^n\), \(n=1,2,\dots\), then the following identities hold \[ \begin{multlined}\sum_{k=-\infty}^\infty\frac{(q/a,q/b,q/c,q/d,q/e)_k}{(a,b,c,d/q,e/q)_k}(abcdeq^{-3})^k\\ =\frac{(q,ab/q,bc/q,ac/q)_\infty}{(a,b,c,abc/q^{2})_\infty}\sum_{k=0}^\infty\frac{(q/a,q/b,q/c,de/q^2)_k}{(q,q^3/abc,d,e)_k}q^k,\end{multlined} \] \[ \begin{multlined}\sum_{k=-\infty}^\infty\frac{(q/a,q/b,q/c,q/d,q/e)_k}{(a,b,c,d/q,e/q)_k}(abcdeq^{-4})^k\\ = \frac{(q,ab/q,bc/q,ac/q)_\infty}{(a,b,c,abc/q^{2})_\infty}\sum_{k=0}^\infty\frac{(q/a,q/b,q/c,de/q^2)_k}{(q,q^3/abc,d,e)_k}q^{2k}.\end{multlined} \]
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