Inversion formulas for elliptic functions. (English) Zbl 1248.11031

Cooper, Shaun

Proc. Lond. Math. Soc. (3) 99, No. 2, 461-483 (2009).

Author’s abstract: 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 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.

Reviewer: Yang Yifan (Hsinchu)

MSC:

11F11	Holomorphic modular forms of integral weight
11F20	Dedekind eta function, Dedekind sums
33D52	Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)

Keywords:

hypergeometric functions; Ramanujan’s inversion formulas; elliptic functions; theta functions; Eisenstein series