Transformation properties for Dyson’s rank function. (English) Zbl 1469.11404
Summary: At the 1987 Ramanujan Centenary meeting Dyson asked for a coherent group-theoretical structure for Ramanujan’s mock theta functions analogous to Hecke’s theory of modular forms. Many of Ramanujan’s mock theta functions can be written in terms of \(R(\zeta ,q)\), where \(R(z,q)\) is the two-variable generating function of Dyson’s rank function and \(\zeta \) is a root of unity. Building on earlier work of G. N. Watson [J. Lond. Math. Soc. 11, 55–80 (1936; JFM 62.0430.02)], S. P. Zwegers [Contemp. Math. 291, 269–277 (2001; Zbl 1044.11029)], B. Gordon and R. J. McIntosh [J. Lond. Math. Soc., II. Ser. 62, No. 2, 321–335 (2000; Zbl 1031.11007)], and motivated by Dyson’s question, Bringmann, Ono, and Rhoades [K. Bringmann and K. Ono, Ann. Math. (2) 171, No. 1, 419–449 (2010; Zbl 1277.11096); K. Bringmann et al., J. Am. Math. Soc. 21, No. 4, 1085–1104 (2008; Zbl 1208.11065)] studied transformation properties of \( R(\zeta ,q)\). In this paper we strengthen and extend the results of Bringmann, Rhoades, and Ono, and the later work of S. Ahlgren and S. Treneer [Acta Arith. 133, No. 3, 267–279 (2008; Zbl 1234.11059)]. As an application we give a new proof of Dyson’s rank conjecture and show that Ramanujan’s Dyson rank identity modulo 5 from the Lost Notebook has an analogue for all primes greater than 3. The proof of this analogue was inspired by recent work of C. Jennings-Shaffer [J. Number Theory 163, 331–358 (2016; Zbl 1408.11103)] on overpartition rank differences \(\bmod 7\).
MSC:
| 11P82 | Analytic theory of partitions |
| 05A19 | Combinatorial identities, bijective combinatorics |
| 11B65 | Binomial coefficients; factorials; \(q\)-identities |
| 11F11 | Holomorphic modular forms of integral weight |
| 11P83 | Partitions; congruences and congruential restrictions |
| 11P84 | Partition identities; identities of Rogers-Ramanujan type |
| 11F37 | Forms of half-integer weight; nonholomorphic modular forms |
| 33D15 | Basic hypergeometric functions in one variable, \({}_r\phi_s\) |
Citations:
Zbl 1044.11029; Zbl 1031.11007; Zbl 1277.11096; Zbl 1208.11065; Zbl 1234.11059; Zbl 1408.11103; JFM 62.0430.02References:
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