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The Septic Iteration

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 [###1#]

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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