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

[math]\displaystyle{ {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]\displaystyle{ 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 − 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:

[math]\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)} }[/math]

References

• Metropolis, N.; Rota, Gian-Carlo (1984), "The cyclotomic identity", in Greene, Curtis, 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., 34, Providence, R.I.: American Mathematical Society, pp. 19–27, ISBN 978-0-8218-5029-9, https://books.google.com/books?id=2axt00oBDEwC&pg=PA19