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Nonic Iterations

In this section we explore nonic (ninth order) iterations. Of course, we may construct a ninth order iteration by iterating the cubic. We have found a nonic iteration which does not seem to come from the cubic. Our nonic iteration is based on a symbolic discovery

that is a nice eta-product:

This identity can be proved easily from results in Ramanujan's notebook. From (3.15), (3.16) we have

by [1, Entry 1(iv), Chapter 20, p. 345,]. In the last step we have used the classical cubic modular equation for the eta-function:

See [7, Corollary 2.5,]. Using (6.5), (6.6)

may be written terms of either

or

(defined in (6.2), (6.3)), which were involved in the cubic iteration. We have

This time we let

where

,

,

are defined in (6.1)--(6.3). We define the following functions:

Using the results of Section 6 it can be shown that

where

. We can also utilize the initial values for the cubic and obtain a nonic iteration.

Theorem 8.1

Define sequences , , and by
where
and where
Then converges nonically to .

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Contents Next: A Sixteenth Order Up: Approximations to via the Previous: The Septic Iteration