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

Research paper by Basil Gordon, Richard J. Mcintosh

Indexed on: 01 Mar '03Published on: 01 Mar '03Published in: The Ramanujan Journal



Abstract

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 e2πir (r rational), there is a theta function Fr(q) with F(q) − Fr(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 = eπiτ. 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.