Inversion formulas for elliptic functions

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.