Cyclotomic identity

Template:Short description In mathematics, the cyclotomic identity states that

$\frac{1}{1-\alpha z}=\prod _{j=1}^{\mathrm{\infty }}{\left(\frac{1}{1-z^j}\right)}^{M(\alpha ,j)}$

where M is Moreau's necklace-counting function,

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

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:

$\prod _{j=1}^{\mathrm{\infty }}{\left(\frac{1}{1-\alpha z^j}\right)}^{M(\beta ,j)}=\prod _{j=1}^{\mathrm{\infty }}{\left(\frac{1}{1-\beta z^j}\right)}^{M(\alpha ,j)}$

References

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