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The H and K Families of Mock Theta Functions

  Published:2011-10-05
 Printed: Aug 2012
  • Richard J. McIntosh,
    Department of Mathematics and Statistics, University of Regina, Regina, SK S4S 0A2
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Abstract

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 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: mock theta function, $q$-series, Appell-Lerch sum, generalized Lambert series mock theta function, $q$-series, Appell-Lerch sum, generalized Lambert series
MSC Classifications: 11B65, 33D15 show english descriptions Binomial coefficients; factorials; $q$-identities [See also 05A10, 05A30]
Basic hypergeometric functions in one variable, ${}_r\phi_s$
11B65 - Binomial coefficients; factorials; $q$-identities [See also 05A10, 05A30]
33D15 - Basic hypergeometric functions in one variable, ${}_r\phi_s$
 

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