New trigonometric sums by sampling theorem

We use a sampling theorem associated with second-order discrete eigenvalue problems to derive some trigonometric identities extending the results of Byrne and Smith [G.J. Byrne, S.J. Smith, Some integer-valued trigonometric sums, Proc. Edinburg Math. Soc. 40 (1997) 393–401]. We derive both integral and non-integral valued trigonometric sums. We give illustrative examples involving representations of the trigonometric sums n k=0 cot2m((2k +1)π/2(2n+1)) and n k=0 tan2m(kπ/(2n + 1)) in an integral-valued polynomial in (2n+1) of degree 2m, m = 1, 2, . . . . © 2007 Elsevier Inc. All rights reserved.