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