* 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...

Recently, Tremblay, Gaboury and Fugère introduced a class of the generalized Bernoulli polynomials (see Tremblay in Appl. Math. Let. 24:1888-1893, 2011). In this paper, we introduce and investigate an extension of the generalized Apostol-Euler polynomials. We state some properties for these polynomials and obtain some relationships between the polynomials and Apostol-Bernoulli polynomials, Stir...

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...

We first introduce a generalization of the Bernoulli polynomials, and consequently of the Bernoulli numbers, starting from suitable generating functions related to a class of Mittag-Leffler functions. Furthermore, multidimensional extensions of the Bernoulli and Appell polynomials are derived generalizing the relevant generating functions, and using the Hermite-Kampé de Fériet (or Gould-Hopper)...

In this work we focus on a sentence retrieval task to present a comparison between Language Modeling based on a multi-variate Bernoulli distribution and Language Modeling based on the popular multinomial models. Nowadays, a view on text generation as a multiple Bernoulli process is not predominant in Language Modeling for Information Retrieval but we show how the characteristics of the task are...

We show that the Akiyama-Tanigawa algorithm and Chen’s variant for computing Bernoulli numbers can be generalized to Carlitz’s q-Bernoulli numbers. We also put these algorithms in the larger context of generalized Euler-Seidel matrices.