New trigonometric sums by sampling theorem

Abstract

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 cot 2m ((2k + 1)π /2(2n + 1)) and n k=0 tan 2m (kπ/(2n + 1)) in an integral-valued polynomial in (2n + 1) of degree 2m, m = 1, 2, . . . .

On an approach for evaluating certain trigonometric character sums using the discrete time heat kernel

“…In [AH18], the authors proved numerous relations by appealing to results in the theory of special functions. In [Ha08], some trigonometric sums were computed by using a discrete form of the sampling theorem associated with certain second-order discrete eigenvalue problems. Each of the above mentioned approaches has proved interesting formulas in specific instances or with certain types of series.…”

Human and automated approaches for finite trigonometric sums

“…Such trigonometric sums occur in a large variety of domains, and are addressed with a large variety of methods: compiling all the pertinent papers in the domain would lead at least to a book (look for example at the long list of references given in [5]). Various methods have been used, some of them in papers cited above: interpolation formulas [46,47,2,33] (to which we can add, e.g., [10,1]...), Ramanujan's theta function [7,32,51], calculus of residues (see, among several other papers, the papers by Cvijović or Cvijović et al cited in [14]; also see [30]), expansions in partial fractions (see, e.g., [13,53,12]), discrete Fourier analysis (see [3]), etc.…”

“…Remark 7 Note that the identities (I, II, III, IV) in Proposition 4 are, up to notation, respectively the identities (3.40, 3.28, 3.14, 3.4) in the paper [33], where they are also generalized.…”

Tangent power sums and their applications

“…Hassan [4] proved these results (see his Theorem 4.3 and formula 4.19), using a sampling theorem associated with the second-order discrete eigenvalue problem.…”

“…In 2002, Chen [1] found formulas for σ(n, p) in case p ≤ 5 as polynomials in n. In 2007-2008, Shevelev [12] and Hassan [4] independently proved the following statements:…”