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| In [[mathematics]], the '''cyclotomic identity''' states that
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| :<math>{1 \over 1-\alpha z}=\prod_{j=1}^\infty\left({1 \over 1-z^j}\right)^{M(\alpha,j)}</math>
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| where ''M'' is [[Moreau's necklace-counting function]],
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| :<math>M(\alpha,n)={1\over n}\sum_{d\,|\,n}\mu\left({n \over d}\right)\alpha^d,</math>
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| and ''μ''(·) is the classic [[Möbius function]] of [[number theory]].
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| The name comes from the denominator, 1 − ''z''<sup> ''j''</sup>, which is the product of [[cyclotomic polynomial]]s.
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| The left hand side of the cyclotomic identity is the [[generating function]] for the free associative algebra on α generators, and the right hand side is the generating function for the [[universal enveloping algebra]] of the [[free Lie algebra]] on α generators. The cyclotomic identity witnesses the fact that these two algebras are isomorphic. | |
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| There is also a symmetric generalization of the cyclotomic identity found by Strehl:
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| :<math>\prod_{j=1}^\infty\left({1 \over 1-\alpha z^j}\right)^{M(\beta,j)}=\prod_{j=1}^\infty\left({1 \over 1-\beta z^j}\right)^{M(\alpha,j)}</math> | |
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| ==References==
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| *{{Citation | last1=Metropolis | first1=N. | last2=Rota | first2=Gian-Carlo | author2-link=Gian-Carlo Rota | editor1-last=Greene | editor1-first=Curtis | editor1-link = Curtis Greene | title=Combinatorics and algebra (Boulder, Colo., 1983). Proceedings of the AMS-IMS-SIAM joint summer research conference held at the University of Colorado, Boulder, Colo., June 5–11, 1983. | url=http://books.google.com/books?id=2axt00oBDEwC&pg=PA19 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Contemp. Math. | isbn=978-0-8218-5029-9 | mr=777692 | year=1984 | volume=34 | chapter=The cyclotomic identity | pages=19–27}}
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| [[Category:Mathematical identities]]
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