Generalizations of mock theta functions and radial limits

Abstract:

In the last letter to Hardy, Ramanujan [Collected Papers, Cambridge Univ. Press, 1927; Reprinted, Chelsea, New York, 1962] introduced seventeen functions defined by $q$-series convergent for $|q|<1$ with a complex variable $q$, and called these functions “mock theta functions”. Subsequently, mock theta functions were widely studied in the literature. In the survey of B. Gordon and R. J. McIntosh [A survey of classical mock theta functions, Partitions, $q$-series, and modular forms, Dev. Math., vol. 23, Springer, New York, 2012, pp. 95–144], they showed that the odd (resp. even) order mock theta functions are related to the function $g_3(x,q)$ (resp. $g_2(x,q)$). These two functions are usually called “universal mock theta functions”. D. R. Hickerson and E. T. Mortenson [Proc. Lond. Math. Soc. (3) 109 (2014), pp. 382–422] expressed all the classical mock theta functions and the two universal mock theta functions in terms of Appell–Lerch sums. In this paper, based on some $q$-series identities, we find four functions, and express them in terms of Appell–Lerch sums. For example, \begin{equation*} 1+(xq^{-1}-x^{-1}q)\sum _{n=0}^{\infty }\frac {(-1;q)_{2n}q^{n}}{(xq^{-1},x^{-1}q;q^2)_{n+1}}=2m(x,q^2,q). \end{equation*} Then we establish some identities related to these functions and the universal mock theta function $g_2(x,q)$. These relations imply that all the classical mock theta functions can be expressed in terms of these four functions. Furthermore, by means of $q$-series identities and some properties of Appell–Lerch sums, we derive four radial limit results related to these functions.