The problem is to find a recurrence relation for the , defined in (4.1). An examination of (3.18) reveals that we need to get in terms of . From (3.14) we see that this is equivalent to finding a relation between , and . Here we have written as a function of q and not r. By using (3.15), it can be shown that is a modular form of weight two on a certain congruence subgroup.

This implies that all three functions , and are modular forms of weight two on a certain congruence subgroup and hence must satisfy an algebraic relation of the form

the dimension of the space of modular forms (above) of weight 2k is asymptotic to some postive constant times k (see [10]). It is also well-known that the number of monomials is asymptotically equal to . Hence there will always be a relation for large enough k. Such a relation can be found and proved symbolically. This is really a linear problem. The q-series expansion of each monomial up to a certain power of q can be easily computed and stored as a column in a matrix. Finding homogeneous relations is then equivalent to finding the nullspace of a certain matrix. Such relations can be proved by verifying them to a high enough power of q using the theory of modular forms . See [7] for more details. We illustrate the case p=2 with a MAPLE session.

> read funcs: > read findhom: > A2:=Aseries(2);

> findhom([A2(q),A2(q),A2(q)],2,100);

# of terms, 22

-----RELATIONS-----of order---,2

We now check the relations to O(q t)

---RELATION----, 1, ---checks to order---