, Volume 7, Issue 1-3, pp 193-222

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

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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 e ir (r rational), there is a theta function F r(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.