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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
Citations:
Zbl 1294.11083
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References:
[1] Andrews, George E.; Berndt, Bruce C., Ramanujan’s lost notebook. Part I, xiv+437 pp. (2005), Springer: New York:Springer · Zbl 1075.11001
[2] Andrews, George E.; Berndt, Bruce C., Ramanujan’s lost notebook. Part II, xii+418 pp. (2009), Springer: New York:Springer · Zbl 1180.11001
[3] [BR95] B.C. Berndt and R.A. Rankin, Ramanujan. Letters and commentary, History of Math., 9 Amer. Math. Soc., Providence, RI and London Math. Soc., London, 1995. · Zbl 0842.01026
[4] Bringmann, Kathrin; Ono, Ken; Rhoades, Robert C., Eulerian series as modular forms, J. Amer. Math. Soc., 21, 4, 1085-1104 (2008) · Zbl 1208.11065
[5] Griffin, Michael; Ono, Ken; Rolen, Larry, Ramanujan’s mock theta functions, Proc. Natl. Acad. Sci. USA, 110, 15, 5765-5768 (2013) · Zbl 1295.11038
[6] [FOR13a] A. Folsom, K. Ono and R.C. Rhoades, Mock theta functions and quantum modular forms, Forum Math. Pi 1 (2013), e2, 27 pp. · Zbl 1294.11083
[7] [FOR13b] A. Folsom, K. Ono and R.C. Rhoades, Ramanujan’s radial limits, Preprint (2012), 12 pp. · Zbl 1359.11064
[8] [Mor13] E. Mortenson, Eulerian series as modular forms revisited, Preprint arXiv:1304.4012 [math.NT] (2013), 6 pp. · Zbl 1334.11036
[9] Ono, Ken, Unearthing the visions of a master: harmonic Maass forms and number theory. Current developments in mathematics, 2008, 347-454 (2009), Int. Press, Somerville, MA · Zbl 1229.11074
[10] Pupyrev, Yu. A., On the linear and algebraic independence of \(q\)-zeta values, Mat. Zametki. Math. Notes, 78 78, 3-4, 563-568 (2005) · Zbl 1160.11338
[11] Zagier, Don, Ramanujan’s mock theta functions and their applications (after Zwegers and Ono-Bringmann), Ast\'erisque, 326, Exp. No. 986, vii-viii, 143-164 (2010) (2009) · Zbl 1198.11046
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