New parameterized mock theta functions and Hecke-type double sums

Abstract

In this paper we express two new mock theta functions with one parameter as the Appell-Lerch sums using the Bailey mechanism. Meanwhile, we also obtain some Hecke-type double sums for some new $q$-series. In addition, we establish the relationships between the mock theta functions and the classical sixth and eighth order mock theta functions.
Furthermore, we give the Hecke-type double sums for the second order mock theta function $D_5(q)$.