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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Some trigonometric identities related to exact covers


Author: John Beebee
Journal: Proc. Amer. Math. Soc. 112 (1991), 329-338
MSC: Primary 11B25; Secondary 11L03
MathSciNet review: 1049133
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Abstract: Sherman K. Stein proves that if $ \sin \pi z = k\prod\limits_{i = 1}^n {\sin } \left( {\pi /{d_i}} \right)\left( {{b_i} - z} \right)$ where the $ {b_i}$ are integers, the $ {d_i}$ are positive integers, $ k$ is a constant, then $ \left\{ {\left( {{d_i}:{b_i}} \right)} \right\}$ is an exact cover. It is shown here that if $ 0 \leq {b_i} < {d_i}$ then $ k = - {2^{n - 1}}$, that the converse is also true, and an analogous formula is conjectured for infinite exact covers. Many well known and lesser known trigonometric and functional identities can be derived from this result and known families of exact covers. A procedure is given for constructing exact covers by induction.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1049133-4
PII: S 0002-9939(1991)1049133-4
Keywords: Exact covering systems, functional identities, trigonometric identities
Article copyright: © Copyright 1991 American Mathematical Society