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The Septic Iteration

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  In this section we examine the case p=7 and obtain an iteration that converges to to 7-th order . The proofs of some of the results of this section have been omitted. They will appear in a later version of this organic paper. Our idea is to mimic the cubic iteration of Section 6. The quadratic form in the cubic case is , which has discriminant -3 and class number .

The case p=7 is the next case to consider since the next odd, negative discriminant with class number 1 is -7. The corresponding quadratic form is . See [9] for a treatment of the classical theory of binary quadratic forms. We have

 

where

  

Also,

 

Equation (7.1) with (7.2), and (7.4) are contained in Entry 5(i) of Chapter 21 of Ramanujan's second notebook [13], [1, p. 467,]. Equation (7.3) is analogous to (6.8) but does not appear in Ramanujan's notebooks. We expect a result like (7.3) to hold since the class number . A result of Legendre's [11, Equation (1), p. 97,] gives an equivalent formulation in terms of L-series:

 

We define the following functions (analogous to the cubic case):

 

and for j=1,2,3

  

In a later version of this paper we will examine these functions under the action of the congruence subgroup , .

The proof of the following is analogous to the cubic case [7, Lemma 2.1, p. 36,]:

    

From (7.9)--(7.12) we easily find that

  

For j=1,2,3 we define

and

  

From (3.14), (7.1) we have

 

Throughout this section q and r are related by

 

If we write the as functions of r then by using the multidimensional analogue [0]

of (3.3) it can be shown that

 

for j=1, 2, 3.

By (7.14), (7.19) we have

  

As noted in Section 4.3 the main problem of constructing a 7-th order iteration is to find a relationship between and . In view of (7.22) and (7.23), we would like to find a relationship between and . Instead we get each of the in terms of , , . This is achieved by solving a certain cubic and taking 7-th roots. We now give some details of this construction. Let

 

Now we define the following functions:

    

Then by using the theory of modular functions it can be shown that

   

Also,

 

Our functions and are related. We find that

 

Hence are the roots of the following cubic equation

 

Now observe that

 

Similarly we find that

 

Thus after solving a certain cubic, and taking 7-th roots we may obtain each of the in terms of , , and . From (3.18) we have

 

Finally, to construct an septic iteration, we need some initial values. Two obvious candidates are , . From (3.10) we have

 

We need to find the values of the for r=1. Fortunately, by (7.21), these coincide with the values of the . So when r=1, and . Thus we can in this case solve equations (7.27) and (7.33) to obtain

 

Thus via (7.34) we find that , , are roots of the cubic

 

Then after solving this cubic we can find the initial values using (7.36). To obtain the septic iteration we need only consider in our identities and equations to obtain the following theorem.

Theorem 7.1

  Define sequences , , , and by

  

where

 

and are the roots of the cubic equation

 

 

where are the roots of the cubic

 

where

  

and where

  

 

 

where

 

and

 

Then converges septically to .


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Contents Next: Nonic Iterations Up: Approximations to via the Previous: The Cubic Iteration