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Ramanujan’s “Lost” Notebook. VII: The sixth order mock theta functions. (English) Zbl 0739.11042

The authors prove 11 identities from Ramanujan’s lost notebook. These involve seven functions defined by \(q\)-series, which are called mock theta functions. Let \[ \begin{aligned} (x)_ n &=(x;q)_ n =\begin{cases} \prod_{i=0}^{n-1}(1-q^ ix), &n\geq 0,\\ \prod_{i=1}^{-n}(1- q^{-i}x)^{-1}, &n<0,\;x\neq q,q^ 2,\dots\hbox{ or } q^{-n}, \end{cases} \\ (x)_ \infty &=(x;q)_ \infty=\prod_{i\geq 0}(1-q^ i x), \hbox{ where } | q| <1, \\ \phi(q) &=\sum_{n\geq 0} {(- 1)^ n q^{n^ 2}(q;q^ 2)_ n \over (-q)_{2n}}, \\ \psi(q) &=\sum_{n\geq 0} {(-1^ n)q^{(n+1)^ 2}(q;q^ 2)_ n \over (- q)_{2n+1}}, \\ j(x,q)&=\sum_ n(-1)^ nq^{\left({x \atop 2}\right)}x^ n, \quad | q|<1. \\ \end{aligned} \] Two of the identities are \[ \begin{aligned} \phi(q^ 9)-\psi(q)-q^{-3}\psi(q^ 9) &={{j(- q^ 3,q^{12})(q^ 6;q^ 6)^ 2_ \infty} \over {j(-q,q^ 4)j(- q^ 9,q^{36})}}, \\ {\psi(\omega q)-\psi(\omega^ 2 q)\over (\omega- \omega^ 2)q} &= {{j(-q,q^ 4)j(-q^ 9,q^{36})(q^ 3;q^ 6)_ \infty} \over {j(-q^ 3,q^{12})}} \\ \end{aligned} \] where \(\omega\) is a primitive cubic root of 1. Results concerning \(\Theta\) functions are proved first. Bailey pair method and constant term method are described next. These are used to derive Hecke type identities for the sixth order mock theta functions. The asymptotics of the functions for \(q\) near a root of unity are also discussed.
Reviewer: M.Cheema (Tucson)

MSC:

11P82 Analytic theory of partitions
11F03 Modular and automorphic functions
05A30 \(q\)-calculus and related topics
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