The object of this paper is to introduce and study the properties unified Apostol-Bernoulli Apostol-Euler polynomials noted by $$\left\{ \mathfrak {V_{n}}(x;\lambda ;\mu )\right\} _{n \ge 0}$$ . We some arithmetic as their connection polynomials. Also, we give derivation integration representations Finally, use umbral calculus approach deduce symmetric identities.

The problem of computing the exact value of ∑ 1/n is so classical (dating back to 1644) that it even has a name of its own: the Basel problem. The city of Basel was the hometown both of the famous brothers Jakob and Johann Bernoulli, who made serious but unsuccessful attempts to solve the problem, and to the prodigious mathematician Leonhard Euler, who found the first solution in 1735, see [3]....

the accuracy and efficiency of the elements proposed by finite element method (fem) considerably depend on the interpolating functions namely shape functions used to formulate the displacement field within the element. in the present study, novel functions, namely basic displacements functions (bdfs), are introduced and exploited for structural analysis of nanobeams using finite element method ...

In this paper, the modified Euler-Bernoulli beam model is presented to examine the influence of surface elasticity and residual surface tension on the critical force of axial buckling of nanotubes in the presence of rotary inertia. An explicit solution is derived for the buckling loads of microscaled Euler beams considering surface effects. The size-dependent buckling behavior of the nanotube d...

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.

The Bernoulli polynomials have important applications in number theory and classical analysis. They appear in the integral representation of differentiable periodic functions since they are employed for approximating such functions in terms of polynomials. They are also used for representing the remainder term of the composite Euler-MacLaurin quadrature rule (see [15]). The Bernoulli numbers [3...