Modular Transformations of Ramanujan's Fifth and Seventh Order Mock Theta Functions

摘要

In his last letter to Hardy, Ramanujan defined 17 functions F(q), where |q| < 1. He called them mock theta functions, because as q radially approaches any point e~(2πir) (r rational), there is a theta function Fr(q) with F(q) - F_r(q) = O(1). In this paper we obtain the transformations of Ramanujan's fifth and seventh order mock theta functions under the modular group generators τ → τ + 1 and τ → -1/τ, where q = e~(πiτ). The transformation formulas are more complex than those of ordinary theta functions. A definition of the order of a mock theta function is also given.