On identities of the Rogers-Ramanujan type (Q5898132)

The author introduces a ``parametrized'' Bailey pair \((\alpha_{d,k,n}(a,q),\beta_{d,k,n}(a,q))\) (whose definition we shall not transcribe here). The cases \((d,k) = (1,2), (2,2), (2,3)\), and \((3,4)\) were studied by Bailey, and in each of these cases the \(\beta_{d,k,n}(a,q)\) is a finite product. Here the author looks at several instances where the \(\beta_{d,k,n}(a,q)\) of the parametrized Bailey pair is (essentially) a single-fold sum, such as \((d,k) = (2,4), (2,1), (3,3), (3,5)\), and \((4,6)\). He lists a number of identities that follow from an application of the Bailey Lemma to these pairs. These identities express a two-fold sum as an infinite product. For example, with the usual \(q\)-series notation we have \[ \sum_{n \geq 0}\sum_{r \geq 0} \frac{q^{n^2+4r^2}}{(q;q^2)_n(q^4;q^4)_r(q^2;q^2)_{n-2r}} = \frac{(q^{12},q^{16},q^{28};q^{28})_{\infty}(-q;q^2)_{\infty}} {(q^2;q^2)_{\infty}}. \] The author also introduces a family of \(q\)-series \(Q_{d,k,i}(a,q)\) which, when \(d=1\), is a family of \(q\)-series employed by Andrews in his proof of Gordon's generalization of the Rogers-Ramanujan identities [\textit{G. E. Andrews}, `Am. J. Math. 88, 844--846 (1966; Zbl 0147.26502)]. The author establishes \(q\)-difference equations for these series, which leads to a ``mild'' extension of the aforementioned result of Gordon. Moreover, these \(q\)-difference equations are also shown to be satisfied by certain series coming from (or closely related to those coming from) an application of the Bailey Lemma to the parametrized Bailey pair. This provides a combinatorial interpretation of such series.