We wish to connect the modular equations of Theorem 1 to . This we contrive via the function alpha defined by:


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.
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