In this paper we investigate generalized poly-Bernoulli numbers. We call them multi-poly-Bernoulli numbers, and we establish a closed formula and a duality property for them.

Bernoulli polynomials play an important role in various expansions and approximation formulas which are useful both in analytic theory of numbers and the classical and the numerical analysis. These polynomials can be defined by various methods depending on the applications. There are six approaches to the theory of Bernoulli polynomials. We prefer here the definition by generating functions giv...

This paper proposes a Poisson multi-Bernoulli mixture (PMBM) filter on the space of sets tree trajectories for multiple target tracking with spawning targets. A trajectory contains all information and its descendants, which appear due to process. Each set branches, where each branch has or one descendants genealogy. For standard dynamic measurement models spawning, posterior is PMBM density, Be...

Random sampling is a popular technique for providing fast approximate query answers, especially in data warehouse environments. Compared to other types of synopses, random sampling bears the advantage of retaining the dataset’s dimensionality; it also associates probabilistic error bounds with the query results. Most of the available sampling techniques focus on table-level sampling, that is, t...

The classical output theorem for the M/M/1 queue, due to Burke (1956), states that the departure process from a stationary M/M/1 queue, in equilibrium, has the same law as the arrivals process, that is, it is a Poisson process. In this paper we show that the associated measure-preserving transformation is metrically isomorphic to a two-sided Bernoulli shift. We also discuss some extensions of B...

* Correspondence: [email protected] Department of Mathematics, Kwangwoon University, Seoul 139701, Republic of Korea Full list of author information is available at the end of the article Abstract Let Pn be the space of polynomials of degree less than or equal to n. In this article, using the Bernoulli basis {B0(x), . . . , Bn(x)} for Pn consisting of Bernoulli polynomials, we investigate s...

The roots of Bernoulli polynomials, Bn(z), when plotted in the complex plane, accumulate around a peculiar H-shaped curve. Karl Dilcher proved in 1987 that, on compact subsets of C, the Bernoulli polynomials asymptotically behave like sine or cosine. Here we establish the asmptotic behavior of Bn(nz), compute the distribution of real roots of Bernoulli polynomials and show that, properly rescal...

We present a computer algebra approach to proving identities on Bernoulli and Euler polynomials by using the extended Zeilberger’s algorithm given by Chen, Hou and Mu. The key idea is to use the contour integral definitions of the Bernoulli and Euler numbers to establish recurrence relations on the integrands. Such recurrence relations have certain parameter free properties which lead to the re...

As a generalization of 2D Bernoulli polynomials, neo-Bernoulli polynomials are introduced from a point of view involving the use of nonexponential generating functions. Their relevant recurrence relations, the differential equations satisfied by them and some other properties are obtained. Especially, we obtain the relationships between them and neo-Hermite polynomials. We also study some other...