Values of the derivatives of the cotangent at rational multiples of pi

By elementary arguments, we deduce closed-form expressions for the values of all derivatives of the cotangent function at rational multiples of pi. These formulae are considerably simpler than Similar ones which were found in a different manner by Kolbig. Also, we show that the values of cot((n))(pi x), n epsilon N, at x = 1/2, 1/3, 2/3, 1/4, 3/4, 1/6 and 5/6 are expressible in terms of the values of the Bernoulli polynomials alone. (C) 2008 Elsevier Ltd. All rights reserved

Closed-form formulae for the derivatives of trigonometric functions at rational multiples of π\pi

In this sequel to our recent note it is shown, in a unified manner, by making use of some basic properties of certain special functions, such as the Hurwitz zeta function, Lerch zeta function and Legendre chi function, that the values of all derivatives of four trigonometric functions at rational multiples of $\pi$ can be expressed in closed form as simple finite sums involving the Bernoulli and Euler polynomials. In addition, some particular cases are considered.Comment: 5 page

Multi-Hamiltonian structures for r-matrix systems

For the rational, elliptic and trigonometric r-matrices, we exhibit the links between three "levels" of Poisson spaces: (a) Some finite-dimensional spaces of matrix-valued holomorphic functions on the complex line; (b) Spaces of spectral curves and sheaves supported on them; (c) Symmetric products of a surface. We have, at each level, a linear space of compatible Poisson structures, and the maps relating the levels are Poisson. This leads in a natural way to Nijenhuis coordinates for these spaces. At level (b), there are Hamiltonian systems on these spaces which are integrable for each Poisson structure in the family, and which are such that the Lagrangian leaves are the intersections of the symplective leaves over the Poisson structures in the family. Specific examples include many of the well-known integrable systems.Comment: 26 pages, Plain Te