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Proceedings of the London Mathematical Society Advance Access originally published online on March 16, 2009
Proceedings of the London Mathematical Society 2009 99(2):461-483; doi:10.1112/plms/pdp007
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© 2009 London Mathematical Society

Inversion formulas for elliptic functions

Shaun Cooper

Institute of Information and Mathematical Sciences
Massey University
Albany Campus
Private Bag 102 904
North Shore Mail Centre
Auckland
New Zealand

Received 19 April 2008.

The aim of this work is to give a unified treatment of the fundamental formulas in Ramanujan's theories of elliptic functions to alternative bases. Our approach relies on well-known results from the theory of theta functions, such as the sum of four squares and sum of eight squares theorems, and their cubic analogues. We prove four inversion theorems, one being classical and the other three belonging to Ramanujan's theories to alternative bases. The connections with iterative means and the corresponding transformation formulas for hypergeometric functions are also established.

2000 Mathematics Subject Classification 11F11, 11F20, 14K25, 33D52, 33E05.


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