The main purpose of this paper is to introduce and investigate a new class of generalized Bernoulli polynomials and Euler polynomials based on the q-integers. The q-analogues of well-known formulas are derived. The q-analogue of the Srivastava–Pintér addition theorem is obtained. We give new identities involving q-Bernstein polynomials.

A series expansion with remainder for functions in a Sobolev space is derived in terms of the classical Bernoulli polynomials, the B-spline scale-space and the continuous wavelet transforms with the derivatives of the standardized B-splines as mother wavelets. In the limit as their orders tend to infinity, the B-splines and their derivatives converge to the Gaussian function and its derivatives...

A systemic study of some familier of the modified q-Euler numbers and polynomials with weight α is presented by using the p-adic qintergralon Zp.The study of these numbers and polynomials yields an interesting q-analoque related to Bernoulli numbers and polynomials. Mathematics Subject Classification: 05A30, 11B68, 11S80, 32P05

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...

In this paper we establish some explicit congruences for Bernoulli polynomials modulo a general positive integer. In particular Voronoi’s and Kummer’s congruences are vastly extended.

The behaviour of real zeroes of the Hurwitz zeta function

The aim of this paper is to construct generating functions for q-beta polynomials. By using these generating functions, we define the q -beta polynomials and also derive some fundamental properties of these polynomials. We give some functional equations and partial differential equations (PDEs) related to these generating functions. By using these equations, we find some identities related to t...