We connect and generalize Matiyasevich’s identity #0102 with Bernoulli numbers and an identity of Candelpergher, Coppo and Delabaere on Ramanujan summation of the divergent series of the infinite sum of the harmonic numbers. The formulae are analytic continuation of Euler sums and lead to new recursion relations for derivatives of Bernoulli numbers. The techniques used are contour integration, ...

We use contour integrals and the Cauchy residue theorem in order to derive several summation formulas, in terms of the higher-order Bernoulli polynomials and the ordinary Bernoulli and Euler polynomials, for a remarkably general family of secant sums. Numerous (known or new) special cases are shown to follow readily from the summation formulas presented in this paper. 2007 Elsevier Inc. All rig...

We introduce a notion of Jacobi-Bernoulli cohomology associated to a semisimplicial Lie algebra (SELA). For an algebraic scheme X over C, we construct a tangent SELA TX and show that the Jacobi-Bernoulli cohomology of TX is related to infinitesimal deformations of X.

The purpose of this article is to present, in a simple way, an analytical approach to special numbers and polynomials. The approach is based on derivative polynomials. The paper is, to some extent, a review article, although it contains some new elements. In particular, it seems that some integral representations for Bernoulli numbers and Bernoulli polynomials are new.

ergodic theory : Bernoulli shifts. There is a class of transformations that play a central role in ergodic theory, namely the Bernoulli shifts. The reason for this is partly because they are in a certain sense the simplest examples of measure-preserving transformations and partly because of their role in applications. Definition of Bernoulli shifts. If we are given positive numbers Pu ' ' ' » P...

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Let {T t} be a smooth flow with positive speed and positive topological entropy on a compact smooth three dimensional manifold, and let μ be an ergodic measure of maximal entropy. We show that either {T t} is Bernoulli, or {T t} is isomorphic to the product of a Bernoulli flow and a rotational flow. Applications are given to Reeb flows.

The Cauchy-type product of two arithmetic functions f and g on nonnegative integers is defined by (f • g)(k) := ∑k m=0 ( k m ) f(m)g(k −m). We explore some algebraic properties of the aforementioned convolution, which is a fundamental characteristic of the identities involving the Bernoulli numbers, the Bernoulli polynomials, the power sums, the sums of products, and so forth.

The relations between the Bernoulli and Eulerian polynomials of higher order and the complete Bell polynomials are found that lead to new identities for the Bernoulli and Eulerian polynomials and numbers of higher order. General form of these identities is considered and generating function for polynomials satisfying this general identity is found.