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
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.
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.
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