We introduce the poly-Cauchy polynomials which generalize the classical Cauchy polynomials, and investigate their arithmetical and combinatorial properties. These polynomials are considered as analogues of the poly-Bernoulli polynomials that generalize the classical Bernoulli polynomials. Moreover, we investigate the zeta functions which interpolate the poly-Cauchy polynomials. The values of th...

The principal aim of this paper is to serve the numerical solution of an integralalgebraic 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 t...

Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties, have been studied in the literature with help generating functions their functional equations. In this paper, we define generalized (p,q)-Bernoulli–Fibonacci (p,q)-Bernoulli–Lucas polynomials numbers by using (p,q)-Bernoulli numbers, unified h(x)-Fibonacci h(x)-Lucas polynomials. We als...

In this paper, we introduce a new class of degenerate Hermite poly-Bernoulli polynomials and give some identities of these polynomials related to the Stirling numbers of the second kind. Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating functions. These results extend some known summations and identities of d...

Following the ideas of L. Carlitz we introduce a generalization of the Bernoulli and Eulerian polynomials of higher order to vectorial index and argument. These polynomials are used for computation of the vector partition function W (s,D), i.e., a number of integer solutions to a linear system x ≥ 0, Dx = s. It is shown that W (s,D) can be expressed through the vector Bernoulli polynomials of h...

In this paper, we further study the generating function involving a variety of special numbers and ploynomials constructed by second author. Applying Mellin transformation to function, define new class zeta type functions, which is related interpolation functions Apostol–Bernoulli polynomials, Bernoulli Euler polynomials. This Hurwitz alternating Lerch function. Furthermore, using these derive ...

The current article focus on the ordinary Bernoulli, Euler and Genocchi numbers and polynomials. It introduces a new approach to obtain identities involving these special polynomials and numbers via generating functions. As an application of the new approach, an easy proof for the main result in [6] is given. Relationships between the Genocchi and the Bernoulli polynomials and numbers are obtai...

We use contour integrals and the Cauchy residue theorem in order to derive several summation formulas, in terms of the higher-order Bernoulli polynomials and the ordinary Bernoulli and Euler polynomials, for a remarkably general family of secant sums. Numerous (known or new) special cases are shown to follow readily from the summation formulas presented in this paper. 2007 Elsevier Inc. All rig...

In this paper, we consider the degenerate multi-poly-Bernoulli numbers and polynomials which are defined by means of multiple polylogarithms versions polynomials. We investigate some properties for those addition, give identities relations multi-poly- Bernoulli