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.

One purpose of this paper is to define the twisted q-Bernoulli numbers by using p-adic invariant integrals on Zp. Finally, we construct the twisted q-zeta function and q-L-series which interpolate the twisted q-Bernoulli numbers.

pollution has become a very serious threat to our environment. monitoring pollution is the rst step toward planning to save the environment. the use of dierential equations, monitoring pollution has become possible. in this paper, a ritz-collocation method is introduced to solve non-linear oscillatory systems such as modelling the pollution of a system of lakes. the method is based upon bernoul...

The principal aim of this paper is to serve the numerical solution of an integral-algebraic equation (IAE) by using the Bernoulli polynomials and the residual correction method. After implementation of our scheme, the main problem would be transformed into a system of algebraic equations such that its solutions are the unknown Bernoulli coefficients. This method gives an analytic solution when ...

In this paper, we study the Fourier series related to higher-order Bernoulli functions and give new identities for higher-order Bernoulli functions which are derived from the Fourier series of them.