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);
# of terms, 22
-----RELATIONS-----of order---,2
---RELATION----, 1, ---checks to order---