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On three theorems of Folsom, Ono and Rhoades. (English) Zbl 1311.11026
The subject is the asymptotic behaviour of one of Ramanujan’s mock theta functions. Consider Dyson’s rank function \(R(w,q)\). For a root of unity \(w \neq 1\) it is a mock theta function multiplied by a power of \(q\). Also consider the Andrews-Garvan crank function \(C(w,q)\), which for a root of unity \(w\) is a modular form up to a power of \(q\). Fix a root of unity \(w\) and let \(q\) tend to another root of unity. Then the asymptotics of \(R(w,q)\) and \(C(w,q)\) are related by a theorem of A. Folsom et al. [Forum Math. Pi 1, Article ID e2, 27 p. (2013; Zbl 1294.11083)]. The special case \(w = -1\) gives the asymptotics of Ramanujan’s order 3 mock theta function \(f(q)\). In the present paper a simple proof for this case is given, just using transformations of \(q\)-series.

11F03 Modular and automorphic functions
11P84 Partition identities; identities of Rogers-Ramanujan type
Zbl 1294.11083
Full Text: DOI arXiv
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