The H and K Family of Mock Theta Functions

McIntosh, Richard J.

Canadian journal of mathematics, Tome 64 (2012) no. 4, pp. 935-960

Résumé

In his last letter to Hardy, Ramanujan defined 17 functions $F\left( q \right),\,\left| q \right|\,<\,1$ , which he called mock $\theta $ -functions. He observed that as $q$ radially approaches any root of unity $\zeta $ at which $F\left( q \right)$ has an exponential singularity, there is a $\theta $ -function ${{T}_{\zeta }}\left( q \right)$ with $F\left( q \right)\,-\,{{T}_{\zeta }}\left( q \right)\,=\,O\left( 1 \right)$ . Since then, other functions have been found that possess this property. These functions are related to a function $H\left( x,\,q \right)$ , where $x$ is usually ${{q}^{r}}$ or ${{e}^{2\pi ir}}$ 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 Appell–Lerch sums (also called generalized Lambert series). Some relations (mock theta “conjectures”) involving mock $\theta $ -functions of even order and $H$ are listed.

Détail

DOI : 10.4153/CJM-2011-066-0

Mots-clés : 11B65, 33D15, mock theta function, q-series, Appell–Lerch sum, generalized Lambert series