Six approaches to the theory of Bernoulli polynomials are known; these are associated with the names of J. Bernoulli [2], L. Euler [4], E. Lucas [8], P. E. Appell [1], A. Hürwitz [6] and D. H. Lehmer [7]. In this note we deal with a new determinantal definition for Bernoulli polynomials recently proposed by F. Costabile [3]; in particular, we emphasize some consequent procedures for automatic c...

A size-dependent Euler–Bernoulli beam model, which accounts for nonlocal stress field, strain gradient field and higher order inertia force field, is derived based on the nonlocal strain gradient theory considering velocity gradient effect. The governing equations and boundary conditions are derived both in dimensional and dimensionless form by employed the Hamilton principle. The analytical so...

the vibration of cable-stayed bridges subjected to the passage of high-speed trains is studied in this article. the moving train includes a number of wagons, each of which is modeled as a four-axle system possessing 48 degrees of freedom. the car model is nonlinear and three-dimensional and includes nonlinear springs and dampers of primary and secondary suspensions, dry friction between differe...

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...

The wheel-rail-sleepers system is simulated as a series of moving point loads on an Euler–Bernoulli beam resting on a visco-elastic half space. This paper concentrates on the railsleepers interaction system (railway system) and the fault detection. The main objective is to mathematically develop and implement a dynamic model of a railway system then the diagnosis of system defects using a Luenb...

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...

* Correspondence: [email protected] Department of Mathematics, Kwangwoon University, Seoul 139701, Republic of Korea Full list of author information is available at the end of the article Abstract Let Pn be the space of polynomials of degree less than or equal to n. In this article, using the Bernoulli basis {B0(x), . . . , Bn(x)} for Pn consisting of Bernoulli polynomials, we investigate s...

We investigate the influences of modelling assumptions on the dynamic behaviour of collapsible channel flows. The elastic wall is modelled in various different ways: as a large strain Bernoulli–Euler beam, as a small strain Timoshenko beam, and as a 2D-solid model derived from a general virtual work approach, using small strain or large strain assumptions. Different inlet boundary conditions ar...