Factorizations that involve Ramanujan's function \(k(q)=r(q)r^2(q^2)\)
Consider the Rogers-Ramanujan continued fraction defined by \[ r= r(q) = \dfrac{q^{1/5}}{1+} \;\dfrac{q}{1+} \;\dfrac{q^2}{1+} \;\dfrac{q^3}{1+ \cdots}, \quad |q|<1. \] It satisfies the well-known identity \[ \dfrac{1}{r} -1 - r = \dfrac{\eta(\tau/5)}{\eta(5\tau)}, \] where \(q=\exp(2\pi i \tau) \) with \(\text{Im } \tau >0 \) and \(\eta(\tau) \) denotes the Dedekind eta-function. The left-hand side has the factorization given by \[ \dfrac{1}{r} -1 - r = \left(\dfrac{1}{\sqrt r} - \left( \dfrac{1-\sqrt 5}{2}\right) \sqrt r \right) \left(\dfrac{1}{\sqrt r} - \left( \dfrac{1+\sqrt 5}{2}\right) \sqrt r \right). \] In the lost notebook, Ramanujan recorded infinite product expansions for each of the factors. The authors give analogues of these results that involve Ramanujan's function \(k = k(q) = r(q) r^2 (q^2). \) More precisely, the function \(k \) satisfies the identities (see [\textit{S. Cooper}, Ramanujan J. 20, No. 3, 311--328 (2009; Zbl 1239.11051)]) \[ \dfrac{1}{k} - k = \dfrac{\eta(2\tau) \eta^5(5\tau)}{\eta(\tau) \eta^5(10\tau)}, \] \[ \dfrac{1}{k} + 1 - k = \dfrac{\eta^4(2\tau) \eta^2(5\tau)}{\eta^2(\tau) \eta^4(10\tau)}, \] \[ \dfrac{1}{k} - 4 - k =\dfrac{\eta^3(\tau) \eta(5\tau)}{\eta(2\tau) \eta^3(10\tau)}. \] In the paper under review, the authors give infinite product expansions for each of the factors that come from the factorization on the left-hand sides.
scientific article; zbMATH DE number 6145903
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