Some integer-valued trigonometric sums

Abstract

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 nth Chebyshev polynomial T.(x) = cos(narccosx), -1 < x < 1.

New trigonometric sums by sampling theorem

“…We note that (3.8) and (3.12) are the sums which are dealt in [5]. In fact the sums (3.7)-(3.13) can be dealt from the view of the work of [5].…”

“…In fact the sums (3.7)-(3.13) can be dealt from the view of the work of [5]. For example these all sums are integral-valued polynomials (note (3.4)-(3.6) as special cases when m = 1, r = 1) and the sums (3.9) and (3.13) can be shown to have the same integralvalued polynomials (1.4) and (1.7), respectively (up to sign in the second case) as we will refer in Section 4.…”

“…Putting S n (θ ) = sin θ , S n (θ ) = sin nθ and letting θ = 0, we get [5] n k=1 (−1) k−1 cot (2k − 1)π 4n = n, n k=1 cot 2 (2k − 1)π 4n = 2n 2 − n, (1.3) respectively. A polynomial p(x) is said to be integral-valued if p(x) is an integer whenever x is an integer.…”

“…A polynomial p(x) is said to be integral-valued if p(x) is an integer whenever x is an integer. In [5] Byrne and Smith introduced generalizations of the integral-valued identities (1.3) as follows. The leading coefficients of (1.4) are given explicitly by…”

Summation formulae on trigonometric functions

“…Berndt [2] has employed the Cauchy residue theorem to treat the trigonometric reciprocity. Similar trigonometric sums have important applications in classical analysis, such as integer-valued problems by Byrne and Smith [3], Dedekind sums by Gessel [8] and Zagier [11], the matrix spectrum by Calogero [4, §2.4.5.3] as well as trigonometric approximation and interpolation in Kress [9,§8.2]. For the parametric trigonometric sums, refer to the most recent works due to Chu [7] and Mohlenkamp and Monzón [10].…”