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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.
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481:{\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)}}
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for the free associative algebra on α generators, and the right hand side is the generating function for the
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140:{\displaystyle {1 \over 1-\alpha z}=\prod _{j=1}^{\infty }\left({1 \over 1-z^{j}}\right)^{M(\alpha ,j)}}
304:
on α generators. The cyclotomic identity witnesses the fact that these two algebras are isomorphic.
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256:{\displaystyle M(\alpha ,n)={1 \over n}\sum _{d\,|\,n}\mu \left({n \over d}\right)\alpha ^{d},}
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There is also a symmetric generalization of the cyclotomic identity found by Strehl:
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Expresses 1/(1-az) as an infinite product using Moreau's necklace-counting function
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The name comes from the denominator, 1 −
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The left hand side of the cyclotomic identity is the
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512:, Contemp. Math., vol. 34, Providence, R.I.:
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502:(1984), "The cyclotomic identity", in
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156:Moreau's necklace-counting function
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514:American Mathematical Society
298:universal enveloping algebra
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285:, which is the product of
548:Mathematical identities
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287:cyclotomic polynomials
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294:generating function
25:cyclotomic identity
516:, pp. 19–27,
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553:Infinite products
523:978-0-8218-5029-9
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272:Möbius function
270:is the classic
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27:states that
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21:mathematics
542:Categories
492:References
465:α
438:β
435:−
416:∞
401:∏
383:β
356:α
353:−
334:∞
319:∏
242:α
220:μ
201:∑
175:α
124:α
97:−
78:∞
63:∏
50:α
47:−
532:0777692
506:(ed.),
300:of the
530:
520:
150:where
23:, the
518:ISBN
278:.
266:and
274:of
154:is
19:In
544::
528:MR
526:,
289:.
158:,
474:)
471:j
468:,
462:(
459:M
454:)
446:j
442:z
432:1
428:1
423:(
411:1
408:=
405:j
397:=
392:)
389:j
386:,
380:(
377:M
372:)
364:j
360:z
350:1
346:1
341:(
329:1
326:=
323:j
283:z
268:μ
251:,
246:d
237:)
232:d
229:n
224:(
215:n
210:|
205:d
195:n
192:1
187:=
184:)
181:n
178:,
172:(
169:M
152:M
133:)
130:j
127:,
121:(
118:M
113:)
105:j
101:z
94:1
90:1
85:(
73:1
70:=
67:j
59:=
53:z
44:1
40:1
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