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 name comes from the denominator, 1 &minus; zj, which is the product of cyclotomic polynomials.

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:
 * $$\prod_{j=1}^\infty\left({1 \over 1-\alpha z^j}\right)^{M(\beta,j)}=\prod_{j=1}^\infty\left({1 \over 1-\beta z^j}\right)^{M(\alpha,j)}$$