Ramanujan's ``Lost'' Notebook. VII: The sixth order mock theta functions (Q1177238)

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.