In this article we prove a result about sets of coefficients of cyclotomic polynomials. We then give corollaries related to flat cyclotomic polynomials and establish the first known infinite family of flat cyclotomic polynomials of order four. We end with some questions related to flat cyclotomic polynomials of order four and five.

The new method for obtaining a variety of extensions of Hermite polynomials is given. As a first example a family of orthogonal polynomial systems which includes the generalized Hermite polynomials is considered. Apparently, either these polynomials satisfy the differential equation of the second order obtained in this work or there is no differential equation of a finite order for these polyno...

The Bernoulli polynomials Bk restricted to 0, 1 and extended by periodicity have nth sine and cosine Fourier coefficients of the formCk/n . In general, the Fourier coefficients of any polynomial restricted to 0, 1 are linear combinations of terms of the form 1/n . If we can make this linear combination explicit for a specific family of polynomials, then by uniqueness of Fourier series, we get a...

We introduce a family of polynomials that generalizes the Bell polynomials, in connection with the combinatorics of the central moments of the Poisson distribution. We show that these polynomials are dual of the Charlier polynomials by the Stirling transform, and we study the resulting combinatorial identities for the number of partitions of a set into subsets of size at least 2.

We introduce two biparametric families of Apostol-Frobenius-Euler polynomials level-$m$. give some algebraic properties, as well other identities which connect these polynomial class with the generalized $\lambda$-Stirling type numbers second kind, Apostol--Bernoulli polynomials, Apostol--Genocchi Apostol--Euler and Jacobi polynomials. Finally, we will show differential properties this new family

We study a family of polynomials in two variables, identifying them as the moments of a two-parameter family of orthogonal polynomials. The coefficient array of these orthogonal polynomials is shown to be an ordinary Riordan array. We express the generating function of the sequence of polynomials under study as a continued fraction, and determine the corresponding Hankel transform. An alternati...

Recently, several Bell based polynomials such as Bernoulli, Euler, Genocchi and Apostol versions were defined investigated. The main aim of this paper is to introduce the general family Appell polynomials, which includes many new members in addition existing ones, investigate their properties including determinantal representation, recurrence relation, derivative formula, shift operators differ...

In this paper, we use the blending functions of Lupaş type (rational) (p, q)-Bernstein operators based on (p, q)-integers for construction of Lupaş (p, q)-Bézier curves (rational curves) and surfaces (rational surfaces) with shape parameters. We study the nature of degree elevation and degree reduction for Lupaş (p, q)-Bézier Bernstein functions. Parametric curves are represented using Lupaş (p...

This paper presents a Web based system for capturing outlines of 2D shapes using Matlab Web Server. From a simple user interface in HTML, any Web user can upload his data and view the results. Cubi c Bezier curve design is used to capture the outlines. In that, outline is divided into curve segments at corner points and curve approximation of each segment is performed by estimation of Bezier co...

Based on the original image of plants leaves, the edge detection method will be used to extract the geometrical morphology of leaves. The characteristic points of the contour can be calculated by Harris method. The characteristic points will be used to construct the cubic Bezier curve fitting of contours of leaves, the leaf vein is generated by L system. Finally, the simulation drawing of plant...