Cyclotomic Identity
In
mathematics
, the
cyclotomic identity
states that
{1 \over 1-\alpha z}=\prod_{j=1}^\infty\left({1 \over 1-z^j}\right)^{M(\alpha,j)}
where
M
is
Moreau's necklace-counting function
M(\alpha,n)={1\over n}\sum_{d\,|\,n}\mu\left({n \over d}\right)\alpha^d
and μ is the classic
Möbius function
of number theory. The denominator on the right, 1 −
z
j
, is a
cyclotomic polynomial
-- hence the name.
Reference
Nicholas Metropolis
&
Gian-Carlo Rota
.
The Cyclotomic Identity.
Reprinted in
Gian-Carlo Rota on Combinatorics
. Birkhäuser. Boston. 1995.
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