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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.

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