In the last letter to Hardy, Ramanujan [Collected Papers, Cambridge Univ. Press, 1927; Reprinted, Chelsea, New York, 1962] introduced seventeen functions defined by
q
q
-series convergent for
|
q
|
>
1
|q|>1
with a complex variable
q
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
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
)
g_3(x,q)
(resp.
g
2
(
x
,
q
)
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
q
-series identities, we find four functions, and express them in terms of Appell–Lerch sums. For example,
1
+
(
x
q
−
1
−
x
−
1
q
)
∑
n
=
0
∞
(
−
1
;
q
)
2
n
q
n
(
x
q
−
1
,
x
−
1
q
;
q
2
)
n
+
1
=
2
m
(
x
,
q
2
,
q
)
.
\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
)
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
q
-series identities and some properties of Appell–Lerch sums, we derive four radial limit results related to these functions.