We wish to connect the
modular equations
of
Theorem 1 to . This we contrive via the function
alpha
defined by:
where
This allows one to rewrite Legendre's equation (5.3) in a one-sided form without
the conjugate variable as
We have suppressed, and will continue to suppress, the k variable. With (5.6)
and (5.7) at hand we can write a q-expansion for , namely,
and we can see that as r tends to infinity tends to zero and
tends to . In fact
The key now is iteratively to calculate . This is the content of the next
theorem.