New mock theta conjectures Part I

In their paper “A survey of classical mock theta functions”, Gordon and McIntosh observed that the classical mock (theta )-functions, including those found by Ramanujan, can be expressed in terms of two ‘universal’ mock (theta )-functions denoted by (g_{2}) and (g_{3}). These identities are known as mock (theta )-conjectures. The fifth- and seventh-order mock (theta )-conjectures were proved by Dean Hickerson. In the survey paper the authors gave mock (theta )-conjectures for the other mock (theta )-functions and referred the proofs to a future paper with this title, listed in their references as [GM4]. The purpose of this paper is to prove these identities for the functions of orders 2 and 3.