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Some integer-valued trigonometric sums

Published online by Cambridge University Press:  20 January 2009

Graeme J. Byrne  and
Simon J. Smith
Graeme J. Byrne
Affiliation:
Department of Mathematics, La Trobe University, Bendigo, P.O. Box 199, Bendigo, Victoria, 3552, Australia
Simon J. Smith
Affiliation:
Department of Mathematics, La Trobe University, Bendigo, P.O. Box 199, Bendigo, Victoria, 3552, Australia
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Abstract

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It is shown that for m = 1,2,3,…, the trigonometric sums and can be represented as integer-valued polynomials in n of degrees 2m – 1 and 2m, respectively. Properties of these polynomials are discussed, and recurrence relations for the coefficients are obtained. The proofs of the results depend on the representations of particular polynomials of degree n – 1 or less as their own Lagrange interpolation polynomials based on the zeros of the nth Chebyshev polynomial Tn(x) = cos(narccos x), -1≤x≤1.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 40 , Issue 2 , June 1997 , pp. 393 - 401
Copyright
Copyright © Edinburgh Mathematical Society 1997

References

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