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The functions

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