We characterize the generalized Chebyshev polynomials of the second kind (Chebyshev-II), and then we provide a closed form of the generalized Chebyshev-II polynomials using the Bernstein basis. These polynomials can be used to describe the approximation of continuous functions by Chebyshev interpolation and Chebyshev series and how to efficiently compute such approximations. We conclude the pap...

We introduce here a generalization of the modified Bernstein polynomials for Jacobi weights using the q-Bernstein basis proposed by G.M. Phillips to generalize classical Bernstein Polynomials. The function is evaluated at points which are in geometric progression in ]0, 1[. Numerous properties of the modified Bernstein Polynomials are extended to their q-analogues: simultaneous approximation, p...

We introduce deformations of Kazhdan-Lusztig elements and degenerate nonsymmetric Macdonald polynomials, both of which form a distinguished basis of the polynomial representation of the maximal parabolic subalgebra of the Hecke algebra. We give explicit integral formula for these polynomials, and explicitly describe the transition matrices between classes of polynomials. We further develop a co...

We consider the condition of orthogonal polynomials, encoded by the coeecients of their three-term recurrence relation, if the measure is given by modiied moments (i.e. integrals of certain polynomials forming a basis). The results concerning various polynomial bases are illustrated with simple examples of generating (possibly shifted) Chebyshev polynomials of rst and second kind.

We give a nonrecursive combinatorial formula for the expansion of a stable Grothendieck polynomial in the basis of stable Grothendieck polynomials for partitions. The proof is based on a generalization of the EdelmanGreene insertion algorithm. This result is applied to prove a number of formulas and properties for K-theoretic quiver polynomials and Grothendieck polynomials. In particular we for...

We introduce here a generalization of the modified Bernstein polynomials for Jacobi weights using the q-Bernstein basis proposed by G.M. Phillips to generalize classical Bernstein Polynomials. The function is evaluated at points which are in geometric progression in ]0, 1[. Numerous properties of the modified Bernstein Polynomials are extended to their q-analogues: simultaneous approximation, p...

s of the talks Alexander Aptekarev, Keldysh Institut, Moscow (Russia) Title:” On a discrete entropy of ortogonal polynomials” Abstract: We shall discuss geometrical meaning of the discrete entropy of the eigenvector basis and particularly eigenvectors formed by orthogonal polynomials. The we present a nice new formula for the discrete entropy of Tchebyshev polynomials. It is joint work with J.S...

Using a new formulation of the Bézout matrix, we construct bivariate matrix polynomials expressed in a tensor-product Lagrange basis. We use these matrix polynomials to solve common tasks in computer-aided geometric design. For example, we show that these bivariate polynomials can serve as stable and efficient implicit representations of plane curves for a variety of curve intersection problems.

We introduce the sequence of generalized Gončarov polynomials, which is a basis for the solutions to the Gončarov interpolation problem with respect to a delta operator. Explicitly, a generalized Gončarov basis is a sequence (tn(x))n≥0 of polynomials defined by the biorthogonality relation εzi(d (tn(x))) = n!δi,n for all i, n ∈ N, where d is a delta operator, Z = (zi)i≥0 a sequence of scalars, ...

The generalized Bernstein basis in the space Πn of polynomials of degree at most n, being an extension of the q-Bernstein basis introduced recently by G.M. Phillips, is given by the formula (see S. Lewanowicz & P. Woźny, BIT 44 (2004), 63–78) Bn i (x;ω| q) := 1 (ω; q)n [ n i ] q x (ωx−1; q)i (x; q)n−i (i = 0, 1, . . . , n). We give explicitly the dual basis functions Dn k (x; a, b, ω| q) for th...