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.