Ramanujan's ``Lost'' Notebook. VI: The mock theta conjectures (Q1123223)

In his last letter to G. H. Hardy, S. Ramanujan spoke of a new class of functions which behaved asymptotically like \(\theta\)-functions but which were not. He called them mock \(\theta\)-functions. Specifically, these are q-series, f(q), convergent for \(| q| <1\) and satisfying: (0) For every root of unity, \(\zeta\), there is a \(\theta\)-function \(\theta_{\zeta}(q)\) such that \(f(q)-\theta_{\zeta}(q)\) is bounded as \(q\to \zeta\) radially. (1) No single \(\theta\)-function works for all \(\zeta\). As Andrews and Hickerson have pointed out, no one has proved that mock \(\theta\)-functions exist. In his letter, Ramanujan listed 17 functions which he proposed as likely candidates. Four of these were described as 3rd order functions, ten as 5th order, and three as 7th order. No one knows precisely what he meant by the order of a mock theta function. \textit{G. N. Watson} [J. Lond. Math. Soc. 11, 55-80 (1936; Zbl 0013.11502)] proved that Ramanujan's 3rd order functions satisfy (0) and are not \(\theta\)-functions, and in [Proc. Lond. Math. Soc., II. Ser. 42, 274-304 (1936; Zbl 0015.30402)] showed that the 5th order functions satisfy (0). The situation rested pretty much at that point until Andrews' discovery of Ramanujan's ``Lost Notebook'' [see Am. Math. Mon. 86, 89-108 (1979; Zbl 0401.01003)] which brought to light ten identities which imply that the 5th order functions are not \(\theta\)-functions. These identities have recently been proven by \textit{D. Hickerson} [Invent. Math. 94, No. 3, 639-660 (1988; Zbl 0661.10059)]. This article is an investigation of those ten identities, proving that they separate into two sets of equivalent identities and investigating their combinatorial implications.