In the paper, the authors establish two identities to express higher order derivatives and integer powers of the generating function of the Chebyshev polynomials of the second kind in terms of integer powers and higher order derivatives of the generating function of the Chebyshev polynomials of the second kind respectively, find an explicit formula and an identity for the Chebyshev polynomials ...

Let (E, ‖·‖E) be a symmetric space and let Y ⊂ X be a nonempty subset. For x ∈ X denote PY (x) = {y ∈ Y : ‖x− y‖ = dist(x, Y )}. Any element y ∈ PY (x) is called a best approximant in Y to x. A nonempty set Y ⊂ X is called proximinal or set of existence if PY (x) 6= ∅ for any x ∈ X. A nonempty set Y is said to be a Chebyshev set if it is proximinal and PY (x) is a singleton for any x ∈ E. A sym...

We describe the development of a scoring function designed to model the hydrophobic effect in protein folding. An optimization technique is used to determine the best functional form of the hydrophobic potential. The scoring function is expanded using the Chebyshev polynomials, for which the coefficients are determined by minimizing the Z-score of native structures in the ensembles of alternate...

Structure constants for the multiplication of Schubert polynomials by Schur symmetric polynomials are related to the enumeration of chains in a new partial order on S 1 , the Grassmaniann Bruhat order. Here we present a monoid M related to this order. We develop a notion of reduced sequences for M and show that M is analogous to the nil-Coxeter monoid for the weak order on S 1 .

In this paper we propose the Bernstein polynomials to achieve the numerical solutions of nonlinear fractional-order chaotic system known by fractional-order Brusselator system. We use operational matrices of fractional integration and multiplication of Bernstein polynomials, which turns the nonlinear fractional-order Brusselator system to a system of algebraic equations. Two illustrative exampl...

The aim of this paper is to study and investigate some new properties of the beta polynomials. Taking derivative of the generating functions for beta type polynomials, we give two partial differential equations (PDEs). By using these PDEs, we derive derivative formulas of the beta type polynomials. In order to construct a matrix representation for the beta polynomials, we firstly show that the ...

The three-dimensional high-order simulation algorithm HOSIM is developed to simulate complex nonlinear and non-Gaussian systems. HOSIM is an alternative to the current MP approaches and it is based upon new high-order spatial connectivity measures, termed high-order spatial cumulants. The HOSIM algorithm implements a sequential simulation process, where local conditional distributions are gener...

The main goal of the present paper is to construct some families of the Carlitz's $q$-Bernoulli polynomials and numbers. We firstly introduce the modified Carlitz's $q$-Bernoulli polynomials and numbers with weight ($_{p}$. We then define the modified degenerate Carlitz's $q$-Bernoulli polynomials and numbers with weight ($alpha ,beta $) and obtain some recurrence relations and other identities...

We present new classes of permutation polynomials over finite fields. If q is the order of a finite field, some polynomials are of form xrf(x(q−1)/d), where d|(q − 1). Other permutation polynomials are related with the trace function. 2000 Mathematics Subject Classification: Primary 11T06.

A four-parameter family of multivariable big q-Jacobi polynomials and a threeparameter family of multivariable little q-Jacobi polynomials are introduced. For both families, full orthogonality is proved with the help of a second-order q-difference operator which is diagonalized by the multivariable polynomials. A link is made between the orthogonality measures and R. Askey’s q-extensions of Sel...