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The H and K Family of Mock Theta Functions

Published online by Cambridge University Press:  20 November 2018

Richard J. McIntosh
Richard J. McIntosh*
Affiliation:
Department of Mathematics and Statistics, University of Regina, Regina, SK S4S 0A2 email: mcintosh@math.uregina.ca
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Abstract

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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.

Keywords

11B6533D15mock theta functionq-seriesAppell–Lerch sumgeneralized Lambert series
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 4 , 01 August 2012 , pp. 935 - 960
Copyright
Copyright © Canadian Mathematical Society 2012

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