A Remarkable eta-Product Identity
                          Michael Somos 11 Nov 2007
                            somos@cis.csuohio.edu
                              (draft version 1)

     Early in 2005 I discovered a remarkable eta-product identity
that was previously known to a very few people implicitly and even
fewer explicitly. In order to state the identity we first define

               y(q) = (1-q)(1-q^2)(1-q^3)(1-q^4) ...

where  |q|<1  is required for convergence. This is Ramanujan's theta
function  f(-q)  and is essentially the Dedekind eta function. For
brevity denote  y(q^n)  by  un . For example, u1 = y(q), u2 = y(q^2),
and so on. Then the following identity holds for  |q|<1

        0 = u1*u12*u15*u20 - u2*u6*u10*u30 + q*u3*u4*u5*u60.

Notice that each term is the product of distinct eta-product factors
with  q^n  allowed. That is, each term is linear in each factor of
y(q^n) which appears. This seems to be the only identity of its kind.
I conjecture this based on my database of thousands of eta-product
identities which I update frequently and which is available upon
request. The identity is related to the McKay-Thompson series of class
60D more details of which are in my frequently updated database on
Monstrous Moonshine. More information about  y(q)  is in my essay
"A Multisection of q-Series" which is on the web.