In this paper, a group shift is an expansive action of Zd on a compact metrizable zero dimensional group by continuous automorphisms. All group shifts factor topologically onto equal-entropy Bernoulli shifts; abelian group shifts factor by continuous group homomorphisms onto canonical equalentropy Bernoulli group shifts; and completely positive entropy abelian group shifts are weakly algebraica...

For every countable group G, a family of isomorphism invariants for measurepreserving G-actions on probability spaces is defined. In the special case in which G is a countable sofic group, a special class of these invariants are computed exactly for Bernoulli systems over G. This leads to a complete classification of Bernoulli systems for many countable groups including all finitely generated l...

We analyze the asymptotic behavior of the Apostol-Bernoulli polynomials Bn(x;λ) in detail. The starting point is their Fourier series on [0, 1] which, it is shown, remains valid as an asymptotic expansion over compact subsets of the complex plane. This is used to determine explicit estimates on the constants in the approximation, and also to analyze oscillatory phenomena which arise in certain ...

In this paper we investigate generalized binomial expansions that arise from two-dimensional sequences satisfying a broad generalization of the triangular recurrence for binomial coefficients. In particular, we present a new combinatorial formula for such sequences in terms of a ’shift by rank’ quasi-expansion based on ordered set partitions. As an application, we give a new proof of Dilcher’s ...

Abstract : We study a model of random electric networks with Bernoulli resistances. In the case of the lattice Z, we show that the point-to-point effective resistance between 0 and a vertex v has a variance of order at most (log |v|) 23 whereas its expected value is of order log |v|, when v goes to infinity. When d 6= 2, expectation and variance are of the same order. Similar results are obtain...

The current article focus on the ordinary Bernoulli, Euler and Genocchi numbers and polynomials. It introduces a new approach to obtain identities involving these special polynomials and numbers via generating functions. As an application of the new approach, an easy proof for the main result in [6] is given. Relationships between the Genocchi and the Bernoulli polynomials and numbers are obtai...

We consider two aspects of generalized Bernoulli polynomials B n (z). One aspect is connected with deening functions instead of polynomials by making the degree n of the polynomial a complex variable. In the second problem we are concerned with the asymptotic behaviour of B n (z) when the degree n tends to innnity.

A construction of new sequences of generalized Bernoulli polynomials of first and second kind is proposed. These sequences share with the classical Bernoulli polynomials many algebraic and number theoretical properties. A class of Euler-type polynomials is also presented. © 2007 Elsevier Inc. All rights reserved.

We investigate Fourier expansions for the Apostol-Bernoulli and Apostol-Euler polynomials using the Lipschitz summation formula and obtain their integral representations. We give some explicit formulas at rational arguments for these polynomials in terms of the Hurwitz zeta function. We also derive the integral representations for the classical Bernoulli and Euler polynomials and related known ...

The main object of this paper is to investigate the Apostol-Bernoulli polynomials and the Apostol-Euler polynomials. We first establish two relationships between the generalized Apostol-Bernoulli and Apostol-Euler polynomials. It can be found that many results obtained before are special cases of these two relationships. Moreover, we have a study on the sums of products of the Apostol-Bernoulli...