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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

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