Hecke-type double sums and the Bailey transform

Hecke-type double sums play a crucial role in proving many identities related to mock theta functions given by Ramanujan. In the literature, the Bailey pair machinery is an efficient tool to derive Hecke-type double sums for mock theta functions. In this paper, by using some Bailey pairs and conjugate Bailey pairs, and then applying the Bailey transform, we establish some trivariate identities which imply the Hecke-type double sums for some classical mock theta functions of orders 3, 6, and 10. Meanwhile, we generalize a bivariate Hecke-type identity due to Garvan.