1. CJM 2011 (vol 64 pp. 935)
 McIntosh, Richard J.

The H and K Families of Mock Theta Functions
In his last letter to Hardy, Ramanujan
defined 17 functions $F(q)$, $q\lt 1$, which he called mock $\theta$functions.
He observed that as $q$ radially approaches any root of unity $\zeta$ at which
$F(q)$ has an exponential singularity, there is a $\theta$function
$T_\zeta(q)$ with $F(q)T_\zeta(q)=O(1)$. Since then, other functions have
been found that possess this property. These functions are related to
a function $H(x,q)$, where $x$ is usually $q^r$ or $e^{2\pi i r}$ for some
rational number $r$. For this reason we refer to $H$ as a ``universal'' mock
$\theta$function. Modular transformations of $H$ give rise to the functions
$K$, $K_1$, $K_2$. The functions $K$ and $K_1$ appear in Ramanujan's lost
notebook. We prove various linear relations between these functions using
AppellLerch sums (also called generalized Lambert series). Some relations
(mock theta ``conjectures'') involving mock $\theta$functions
of even order and $H$ are listed.
Keywords:mock theta function, $q$series, AppellLerch sum, generalized Lambert series Categories:11B65, 33D15 
