Four identities for third order mock theta functions. (English) Zbl 1455.11136
The authors provide completely new proofs for the four identities of the third order mock theta functions (using properties of \(q\)-series), which were recorded by S. Ramanujan [The Lost Notebook and other unpublished papers. With an introduction by George E. Andrews. New Delhi: Narosa Publishing House; Berlin (FRG): Springer-Verlag (1988; Zbl 0639.01023)] in its page no. 2 [Entries 1.1 and 1.2] and page no. 17 [Entries 1.3 and 1.4]. However, these identities were first proved by H. Yesilyurt [Adv. Math. 190, No. 2, 278–299 (2005; Zbl 1106.11013)] with the help of a famous lemma given by [A. O. L. Atkin and P. Swinnerton-Dyer, Proc. Lond. Math. Soc. (3) 4, 84–106 (1954; Zbl 0055.03805)]. Please note that the third order mock theta functions are intimately connected with ranks of partitions. The proofs of Entries 1.1-1.3 are not difficult but the proof of Entry 1.4 is considerably more difficult, which is based upon a 2-dissection for two special cases of the rank generating function \(G(z,q)\), when \(z=i\) and \(z\) is a primitive eight root of unity. These 2-dissections of the rank, with their immediate consequences, comprise a second major focus of this paper. The complete proof of Entry 1.4 is given in four parts which are spread within sixteen pages. Some immediate consequences of the results on the 2-dissection of the rank function \(G(i,q)\) and a primitive eight root of unity related to ranks and mock theta functions are also discussed. The proofs are nicely presented and clearly understandable.
Reviewer: M. P. Chaudhary (New Delhi)
MSC:
| 11P83 | Partitions; congruences and congruential restrictions |
| 33D15 | Basic hypergeometric functions in one variable, \({}_r\phi_s\) |
References:
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