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The functions

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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Contents Next: The Symbolic Search Up: Approximations to via the Previous: The function