In this paper, we study the asymptotics of the discrete Chebyshev polynomials tn(z,N) as the degree grows to infinity. Global asymptotic formulas are obtained as n → ∞, when the ratio of the parameters n/N = c is a constant in the interval (0, 1). Our method is based on a modified version of the Riemann-Hilbert approach first introduced by Deift and Zhou.

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...

In this article, a method is presented for transforming the singular Lippmann-Schwinger integral equation to a matrix algebraic equation. This method of computing the matrix elements of the reaction and transition operators is used on the real axis and on the complex plane, respectively. By specifying the elements value of the reaction and transition matrix on the energy-shell, both phase shift...

in this paper we introduce a type of fractional-order polynomials basedon the classical chebyshev polynomials of the second kind (fcss). also we construct the operationalmatrix of fractional derivative of order $ gamma $ in the caputo for fcss and show that this matrix with the tau method are utilized to reduce the solution of some fractional-order differential equations.

We study the problem of reconstructing a sparse polynomial in a basis of Chebyshev polynomials (Chebyshev basis in short) from given samples on a Chebyshev grid of [−1, 1]. A polynomial is called M -sparse in a Chebyshev basis, if it can be represented by a linear combination of M Chebyshev polynomials. For a polynomial with known and unknown Chebyshev sparsity, respectively, we present efficie...

In advancing our prior work on a unified theory for pseudospectral (PS) optimal control, we present new results for PS methods over arbitrary grids. These results provide a way to compare performances among different PS methods and suggest guidelines to choose the proper grids and discretization approaches for solving optimal control problems. The new unified ideas reveal hidden properties of d...

The Chebyshev or L∞ estimator minimizes the maximum absolute residual and is useful in situations where the error distribution has bounded support. In this paper, we derive the asymptotic distribution of this estimator in cases where the error distribution has bounded and unbounded support. We also consider the asymptotics of set-membership estimators such as the Chebyshev centre and maximum in...

using monthly test day records the genetic parameters of iranian holstein cattle in first lactation were studied. data of 277400 test-day milk records from 65320 cows and 2210 sires were analyzed by an animal random regression model using restricted maximum likelihood methodology. the model included herd-test-date, interaction between year-season of calving, days in milk (linear and quadratic) ...

In this paper, summation formulae for the 2-variable Legendre polynomials in terms of certain multi-variable special polynomials are derived. Several summation formulae for the classical Legendre polynomials are also obtained as applications. Further, Hermite-Legendre polynomials are introduced and summation formulae for these polynomials are also established.