If and

''will surely open a whole new field of analysis.''

** Equivalent Modular Parameterization**

This is equivalent to the identities

and where

These are modular forms. So for example

** A Cubic Analogue of the AGM**

Let

The convergence is cubic.

** Proof**: The proof is opaque. It works because
satisfies

In the above notation.

where is, as above, with, Taking

** Equivalent Modular Parameterization**
If

See * Some cubic modular identities of Ramanujan* by J. Borwein, P. Borwein & F. Garvan
in Trans. A.M.S. ** 343** (1994) 35-47.

** Some Explanations**

where

Let

Then

andIf we can compute iteratively we can compute and by a second linked iteration.

There are four particularly interesting cases.

Inverting the ratios gives elliptic modular functions. Which reduces much of this to an algebraic theory.

Finding and proving these iterations can (at least in principal) be effected entirely computationally.

** The Quadratic s=1/4 Iteration**

Let and Then the common limit is