Further new identities of the Rogers-Ramanujan type
Unfortunately, this is a not a good paper. The author claims to have derived Rogers-Ramanujan identities for the moduli \(6s\), \(10s\), \(13s\), \(14s\), \(18s\), \(22s\) and \(30s\), with \(s\) a nonnegative integer. Besides a very large number of typos (few identities are correctly stated) the claims made by the author are mostly trivial. Take as example his identities for the moduli \(18s\). With the standard notation \((a;q)_n=(1-a)\dots(1-aq^{n-1})\) they read \[ \frac{(q^s;q^s)_{\infty}}{(q;q)_{\infty}} \sum_{n,k=0}^{\infty} \frac{q^{s(n+k)^2+ks(3k-1)/2}} {(q^s;q^s)_n(q^s;q^s)_k(q^s;q^{2s})_k} =\prod_{n=1, n\not\equiv 0,\pm 8s \pmod {18s}}^{\infty} \frac{1}{1-q^n} \] (stated in the paper with the \(k\) missing in \((3k-1)\) and the \(0\) missing in \(0,\pm 8s\)). It is of course nonsense to view this as a series of identities for the moduli \(18s\). Indeed, replacing \(q^s\) by \(q\) the above identity is really nothing but a complicated way of writing the modulus \(18\) identity \[ \sum_{n,k=0}^{\infty} \frac{q^{(n+k)^2+k(3k-1)/2}} {(q;q)_n(q;q)_k(q;q^2)_k} =\prod_{n=1, n\not\equiv 0,\pm 8 \pmod {18}}^{\infty} \frac{1}{1-q^n}. \] Identities like this are an easy exercise for anyone familiar with the Bailey lemma.
scientific article; zbMATH DE number 1786056