Equivalent Modular Parameterization
This is equivalent to the identities
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.
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
Let
Then
If 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