cyclotomic identity

{{short description|Expresses 1/(1-az) as an infinite product using Moreau's necklace-counting function}}

In mathematics, the cyclotomic identity states that

:$\{1\; \backslash over\; 1-\backslash alpha\; z\}=\backslash prod\_\{j=1\}^\backslash infty\backslash left(\{1\; \backslash over\; 1-z^j\}\backslash right)^\{M(\backslash alpha,j)\}$

where M is Moreau's necklace-counting function,

:$M(\backslash alpha,n)=\{1\backslash over\; n\}\backslash sum\_\{d\backslash ,|\backslash ,n\}\backslash mu\backslash left(\{n\; \backslash over\; d\}\backslash right)\backslash alpha^d,$

and μ is the classic Möbius function of number theory.

The name comes from the denominator, 1 − z ^j, 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:

:$\backslash prod\_\{j=1\}^\backslash infty\backslash left(\{1\; \backslash over\; 1-\backslash alpha\; z^j\}\backslash right)^\{M(\backslash beta,j)\}=\backslash prod\_\{j=1\}^\backslash infty\backslash left(\{1\; \backslash over\; 1-\backslash beta\; z^j\}\backslash right)^\{M(\backslash alpha,j)\}$