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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

where
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);
`

* # of terms, 22*

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

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

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