SOME INTEGER-VALUED TRIGONOMETRIC SUMS

It is shown that for m = 1,2,3,..., the trigonometric sums Sigma(k=1)( m)(-1)(k-1)cot(2m-1) ((2k - 1)pi/4n) and Sigma(k=1)(n) cot(2m)((2k - 1 )pi/4n) can be represented as integer-valued polynomials in n of degre es 2m - 1 and 2m, respectively. Properties of these polynomials are di scussed, and recurrence relations for the coefficients are obtained. T he proofs of the results depend on the representations of particular p olynomials of degree n - 1 or less as their own Lagrange interpolation polynomials based on the zeros of the nth Chebyshev polynomial T-n(x) = cos(n arccos x), -1 less than or equal to x less than or equal to 1 .