Cyclotomic identity

In mathematics, the cyclotomic identity states that

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

where M is Moreau's necklace-counting function,

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

and μ is the classic Möbius function of number theory.

The name comes from the denominator, 1 − 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:

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

References

• Metropolis, N.; Rota, Gian-Carlo (1984), "The cyclotomic identity", in Greene, Curtis (ed.), 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., Contemp. Math., vol. 34, Providence, R.I.: American Mathematical Society, pp. 19–27, ISBN 978-0-8218-5029-9, MR 0777692