  Motivated by the works of Liu, we provide a unified approach to find
Appell-Lerch series and Hecke-type series representations for mock theta
functions. We establish a number of parameterized identities with two
parameters $a$ and $b$. Specializing the choices of $(a,b)$, we not only give
various known and new representations for the mock theta functions of orders 2,
3, 5, 6 and 8, but also present many other interesting identities. We find that
some mock theta functions of different orders are related to each other, in the
sense that their representations can be deduced from the same
$(a,b)$-parameterized identity. Furthermore, we introduce the concept of false
Appell-Lerch series. We then express the Appell-Lerch series, false
Appell-Lerch series and Hecke-type series in this paper using the building
blocks $m(x,q,z)$ and $f_{a,b,c}(x,y,q)$ introduced by Hickerson and Mortenson,
as well as $\overline{m}(x,q,z)$ and $\overline{f}_{a,b,c}(x,y,q)$ introduced
in this paper. We also show the equivalences of our new representations for
several mock theta functions and the known representations.
