Cranks and dissections in Ramanujan's lost notebook
Employing the standard \(q\)-series notation, let \(F_a(q)\) be the crank generating function \[ F_a(q) = \frac{(q)_{\infty}}{(aq,q/a)_{\infty}}. \] In Ramanujan's lost notebook there appear two congruences for \(F_a(q)\), one of which we display here: \[ F_a(\sqrt{q}) \equiv \frac{f(-q^3,-q^5)}{(-q^2;q^2)_{\infty}} + \left(a-1+\frac{1}{a}\right) \sqrt{q} \frac{f(-q,-q^7)}{(-q^2;q^2)_{\infty}} \pmod{a^2 + a^{-2}}. \] The function \(f(a,b)\) is Ramanujan's theta function, \[ f(a,b) = \sum_{n \in \mathbb{Z}} a^{n(n+1)/2}b^{n(n-1)/2}. \] In this paper the authors prove Ramanujan's two congruences for \(F_a(q)\) as well as three others with a similar flavor. In each case two different proofs are given. In the conclusion, the authors explain an elegant observation, communicated to them by Frank Garvan, which shows that the congruences for \(F_a(q)\) are actually equivalent to some of Garvan's own identities involving the crank generating function [\textit{F. G. Garvan}, Trans. Am. Math. Soc. 305, No. 1, 47--77 (1988; Zbl 0641.10009) and Trans. Am. Math. Soc. 322, No. 1, 79--94 (1990; Zbl 0717.11042)].
scientific article; zbMATH DE number 2136714
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