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In this section we construct an infinite family of
functions
, where p is any integer greater than 1.
Theoretically it possible to find an update to any order; ie. an equation
relating
with
. We will find that
this relation is particularly nice when N=p. This will give rise to p-th
order iterations with a nice form. Our functions are constructed
from the Dedekind eta function instead of the theta functions.
Let
(with
). As usual the Dedekind
eta function is defined as

Then

See [12, p. 121,] for a proof.
Now for p>1 (a positive integer) we define

where
and
.
It should be noted that the functions
and
occured
naturally in the Borwein-Borwein cubic iteration [6],
[7].
Define

where

Here
.
From (3.2) we have

and

which follows from (3.3).
The definition of
was chosen so that it had a form analogous to that of (2.4)
and that it satisfied a transformation like (3.9) below.
Using (3.3) and (3.8) it is not hard to show that

Substituting r=1 gives

Since
as
we see that

Theorem 3.1
let
be fixed. We have

where

and

Further

where


[Proof of Theorem 3.1]
When N=p, the function
has a nice form

so that

The proof of (3.17) follows easily from (3.2), (3.4)
and (3.15). From (3.8) and (3.14) we have

By using (3.18) and (3.19) we find that (3.9)
and (3.18) give rise to two equations involving
and
. These equations may be solved easily to yield


Contents
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Previous: The function