Some integer-valued trigonometric sums

Graeme J. Byrne, Simon J. Smith
1997 Proceedings of the Edinburgh Mathematical Society  
It is shown that for m = l , 2 , 3 the trigonometric sums ELi(-')* ' c o t^'^f c -\)n/4n) and 5Z*=i cot 2 "^^ -l)7i/4n) 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
more » ... n the zeros of the nth Chebyshev polynomial T.(x) = cos(narccosx), -1 < x < 1.
doi:10.1017/s001309150002383x fatcat:yfmjbjthyvci5l62ovgg5v2lu4