Some integer-valued trigonometric sums

Source abstract

It is shown that for m = 1,2,3,…, the trigonometric sums and can be represented as integer-valued polynomials in n of degrees 2 m – 1 and 2 m , 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 n th Chebyshev polynomial T n (x) = cos( n arccos x ), -1≤ x ≤1.