Generalizations of mock theta functions and radial limits

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, [GRAPHICS] 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.