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

The Ramanujan Journal - Tập 7 - Trang 193-222 - 2003
Basil Gordon1, Richard J. Mcintosh2
1Department of Mathematics, University of California, Los Angeles
2Department of Mathematics and Statistics, University of Regina, Regina, Canada

Tóm tắt

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 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.

Tài liệu tham khảo

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The Ramanujan Journal

Tập 7

193-222

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