Some identities with applications

Abstract. For positive integers \(r\) computable formulas for the sums of the doubly infinite series \(\sum_{k\in{\mathbb Z}}1/(z-k\pi)^r\) and \(\sum_{k\in{\mathbb Z}}(-1)^k/(z-k\pi)^r\) will be presented. As applications exact lower and upper bounds for the derivatives of the functions \(\cot\) and \(1/\sin\) will be also shown.