Exact evaluations of finite trigonometric sums by sampling theorems

Basic trigonometric power sums with applications

“…In addition, by introducing the duplication formula for the gamma function, viz., No. 8.335 (1) in [16], we find that the first term on the rhs reduces to the binomial series for 1/ √ 1 − z, while we introduce the definition for σ k (n) or (4.3) into the second term on the rhs. Consequently, we arrive at…”

“…A typical example is the finite sum of powers of the contangent studied by Berndt and Yeap [3], Here, B j denotes the Bernoulli number with index j 0. As described in Appendix A, Berndt and Yeap use contour integration to derive this result, although more recently it has been studied with the aid of sampling theorems [1]. Unfortunately, (1.2) is, if not incorrect, confusing or misleading because it states that the j i cannot equal zero.…”

Discrete determinants and the Gel’fand–Yaglom formula

“…which generalises (11). This identity is an example of a general class of identities discussed in the interesting work by Annaby and Asharabi, [27], where other references can be found.…”

“…which agrees with an expression in [22] for the continuum case. Related is the limit of the simplest Euler-Rayleigh eigenvalue sum, (27). Reverting to physical quantities,…”

The Trace Method for Cotangent Sums

“…We will see later that even for noninteger values of cot α the sum may evaluate to an integer, e.g., for n " 2 and cot α " 1 2 , Lucas numbers appear, see (5.1) below. (2) The second part of Theorem 3.1 could be seen as a very special case the BMV conjecture [40]: if A and B are positive semi-definite matrices, then for all positive integers m, the polynomial in t, TrpA`tBq m , has only non-negative coefficients. The proof above shows that the assertion is also true whenever A is an orthogonal projection of rank one and B is a positive or antisymmetric self-adjoint matrix.…”