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Inversion formulas for elliptic functions
Author(s) -
Cooper Shaun
Publication year - 2009
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdp007
Subject(s) - mathematics , ramanujan's sum , hypergeometric function , inversion (geology) , ramanujan theta function , elliptic function , elliptic integral , pure mathematics , algebra over a field , ramanujan tau function , theta function , mathematical analysis , paleontology , structural basin , biology
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.