and Applied Analysis 3 we derive some interesting identities and relations on the modified q-Bernoulli numbers and polynomials. 2. The Modified q-Bernoulli Numbers and Polynomials with Weight α In this section, we assume α ∈ Q. Now, we define the modified q-Bernoulli numbers with weight α B̃ α n,q as follows:

We investigate some algorithms that produce Bernoulli, Euler and Genocchi polynomials. We also give closed formulas for Bernoulli, Euler and Genocchi polynomials in terms of weighted Stirling numbers of the second kind, which are extensions of known formulas for Bernoulli, Euler and Genocchi numbers involving Stirling numbers of the second kind.

We consider infinitely convolved Bernoulli measures (or simply Bernoulli convolutions) related to the β-numeration. A matrix decomposition of these measures is obtained in the case when β is a PV number. We also determine their Gibbs properties for β being a multi-nacci number, which makes the multifractal analysis of the corresponding Bernoulli convolution possible.

In the paper, the authors recall some known determinantal expressions in terms of the Hessenberg determinants for the Bernoulli numbers and polynomials, find alternative determinantal expressions in terms of the Hessenberg determinants for the Bernoulli numbers and polynomials, and present several new recurrence relations for the Bernoulli numbers and polynomials.

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...