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A Remarkable eta-product Identity \\
Michael Somos 11 Dec 2014 \\
ms639@georgetown.edu \\
(draft version 9)
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Early in 2005 I discovered a remarkable eta-product identity that
was known most probably to a very few people implicitly and even
fewer explicitly. In order to state the identity we first define the
infinite product
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$$ h(q) := q^{(1/24)}(1-q)(1-q^2)(1-q^3)(1-q^4) ... $$
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where $|q|<1$ is required for convergence. This is the Dedekind eta
function defined in terms of $q$. For brevity let $e_n := h(q^n)$.
For example, $e_1 = h(q), e_2 = h(q^2)$, and so on. The following
level 60 three term identity holds
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$$ e_2 e_6 e_{10} e_{30} = e_1 e_{12} e_{15} e_{20} + e_3 e_4 e_5 e_{60}. $$
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Notice that each term is the product of distinct eta-function factors.
That is, each term is linear in each factor of $h(q^n)$ which appears.
This seems to be the only identity of its kind. More precisely,
here is my conjecture :
Given a finite set $S$ of positive integers, define the eta-product $E(S)$
to be the product over all $n$ in $S$ of $h(q^n)$. There exists a unique
triple $(S_1,S_2,S_3)$ such that $E(S_1) = E(S_2) + E(S_3)$ where the three
sets are pairwise disjoint and the GCD of the union is 1. That triple is
$(\{2,6,10,30\}, \{1,12,15,20\}, \{3,4,5,60\})$. Of course, $S_2$ and $S_3$
can be swapped giving the same identity without affecting uniqueness.
I conjecture this based on my database of thousands of eta-product
identities which available at URL ``http://eta.math.georgetown.edu/''. As
evidence I have used systematic search by computer program up to level
250 and have found no other three term identity like this one. However
I found a similar level 210 four term identity which is as follows
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$$ e_1 e_{30} e_{35} e_{42} + e_3 e_{10} e_{14} e_{105} =
e_2 e_{15} e_{21} e_{70} + e_5 e_6 e_7 e_{210}. $$
If we allow non disjoint sets and a numerical coefficient of 2 then
there is a level 30 four term identity which is as follows
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$$ e_1 e_2 e_{15} e_{30} + e_3 e_5 e_6 e_{10} =
e_1 e_3 e_5 e_{15} + 2 e_2 e_6 e_{10} e_{30}. $$
My unique three term identity is related to the McKay-Thompson series
of class 60D for which see sequences A058728 and A143751 in Neil Sloane's
OEIS and more details of which are in my less frequently updated database
on Monstrous Moonshine also available upon request. More information about
$h(q)$ is in my essay ``A Multisection of $q$-Series'' which is available
on the web.
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