In this paper, we investigate the zeta function

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.

A new class of generalized Laguerre-based poly-Bernoulli polynomials are discussed with an attempt to generate new and interesting identities, some are in relation with Stirling number of the second kind. Different analytical means and generating function method is incorporated to derive implicit summation formulae and symmetry identities for generalized Laguerre poly-Bernoulli polynomials. It ...

The purpose of this article is to present, in a simple way, an analytical approach to special numbers and polynomials. The approach is based on derivative polynomials. The paper is, to some extent, a review article, although it contains some new elements. In particular, it seems that some integral representations for Bernoulli numbers and Bernoulli polynomials are new.

A construction of new sequences of generalized Bernoulli polynomials of first and second kind is proposed. These sequences share with the classical Bernoulli polynomials many algebraic and number theoretical properties. A class of Euler-type polynomials is also presented. © 2007 Elsevier Inc. All rights reserved.

We investigate Fourier expansions for the Apostol-Bernoulli and Apostol-Euler polynomials using the Lipschitz summation formula and obtain their integral representations. We give some explicit formulas at rational arguments for these polynomials in terms of the Hurwitz zeta function. We also derive the integral representations for the classical Bernoulli and Euler polynomials and related known ...

K e y w o r d s B e r n o u l l i polynomials, Euler polynomials, Generating functions, Bernoulli numbers, Euler numbers, Addition theorem, Multiplication theorem~ Generalized Bernoulli polynomials and numbers, Generalized Euler polynomials and numbers. 1. I N T R O D U C T I O N T h e classical Bernoulli polynomials Bn(x) and the classical Euler polynomials En(x) are usual ly def ined by m e a...

By a symbolic method, we introduce multivariate Bernoulli and Euler polynomials as powers of polynomials whose coefficients involve multivariate Lévy processes. Many properties of these polynomials are stated straightforwardly thanks to this representation, which could be easily implemented in any symbolic manipulation system. A very simple relation between these two families of multivariate po...