Editing Cyclotomic identity (section)

{{short description|Expresses 1/(1-az) as an infinite product using Moreau's necklace-counting function}}
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 name comes from the denominator, 1&nbsp;&minus;&nbsp;''z''<sup>&nbsp;''j''</sup>, which is the product of [[cyclotomic polynomial]]s.

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. 

There is also a symmetric generalization of the cyclotomic identity found by Strehl:
:<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>