Hecke-type double sums and mock theta functions

Mock theta functions were first introduced by Ramanujan in his last letter to Hardy. In the literature, Hecke-type double sums for the classical mock theta functions play an important role in proving mock theta conjectures. In this paper, with the aid of some $q$-series identities, we establish Hecke-type double sums for some new mock theta functions. All the Hecke-type double sums are represented in the form of $f_{a,b,c}(x,y,q)$ which is defined by Hickerson and Mortenson. Moreover, we express these functions in terms of Appell–Lerch sums and theta functions.