Kalikow proved that the [T, T−1] transformation is not isomorphic to a Bernoulli shift [3]. We show that the scenery factor of the [T, T−1] transformation is not isomorphic to a Bernoulli shift. Moreover we show that it is not Kakutani equivalent to a Bernoulli shift.

Using the Padé approximation of the exponential function, we obtain recurrence relations between Apostol-Bernoulli and between Apostol-Euler polynomials. As applications, we derive some new lacunary recurrence relations for Bernoulli and Euler polynomials with gap of length 4 and lacunary relations for Bernoulli and Euler numbers with gap of length 6.

Six approaches to the theory of Bernoulli polynomials are known; these are associated with the names of J. Bernoulli [2], L. Euler [4], E. Lucas [8], P. E. Appell [1], A. Hürwitz [6] and D. H. Lehmer [7]. In this note we deal with a new determinantal definition for Bernoulli polynomials recently proposed by F. Costabile [3]; in particular, we emphasize some consequent procedures for automatic c...

We continue the study of Rokhlin entropy, an isomorphism invariant for p.m.p. actions of countable groups introduced in the previous paper. We prove that every free ergodic action with finite Rokhlin entropy admits generating partitions which are almost Bernoulli, strengthening the theorem of Abért–Weiss that all free actions weakly contain Bernoulli shifts. We then use this result to study the...

Abstract. In [18], the new q-extension of Bernoulli polynomials and generalized Bernoulli numbers attached to χ were constructed by using p-adic invariant integral on Zp. In this paper we construct the new q-extension of generalized Bernoulli polynomials attached to χ due to author and derive the existence of a specific p-adic interpolation function which interpolate the q-extension of generali...

Simulation results suggest that a maximum size matching (MSM) algorithm will lead to 100% throughput for uniform Bernoulli i.i.d traffic. Previous analysis on the throughput of MSM algorithms enforce deterministic constraints on the joint arrival traffic. In this paper we explore MSM algorithms under Bernoulli arrivals. We show that when the arrival traffic is admissible and cells are scheduled...

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...

A class of identities satisfied by both Bernoulli and Euler polynomials is established. Recurrence relations for Bernoulli and Euler numbers are derived.

The Bernoulli polynomials Bk restricted to 0, 1 and extended by periodicity have nth sine and cosine Fourier coefficients of the formCk/n . In general, the Fourier coefficients of any polynomial restricted to 0, 1 are linear combinations of terms of the form 1/n . If we can make this linear combination explicit for a specific family of polynomials, then by uniqueness of Fourier series, we get a...

A sequence of random variables, each taking values 0 or 1, is called a Bernoulli sequence. We say that a string of length d occurs in a Bernoulli sequence if a success is followed by exactly (d − 1) failures before the next success. The counts of such d-strings are of interest, and in specific independent Bernoulli sequences are known to correspond to asymptotic d-cycle counts in random permuta...