Sufficient conditions have been recently given for a classs of ergodic maps of an interval onto itself: I = [0, 1] ⊂ R1 → I and its associated binary function to generate a sequence of independent and idetically distributed (i.i.d.) random variables. Jacobian elliptic Chebyshev map, its derivative and second derivative induce Jacobian elliptic space curve. A mapping of the space curve onto itse...

We study the tangent analogues tan(n arctanx) of the Chebyshev polynomials from an algebraic viewpoint. They are rational functions of a pleasant form and enjoy several noteworthy properties: a useful composition law, their numerators pn(x) split into the minimal polynomials of the numbers tan kπ/n, they define the elements of the Galois groups of these minimal polynomials, and their algebraic ...

Following an approach originally due to Mahler and sharpened by Chudnovsky, we develop an explicit version of the multi-dimensional \hy-pergeometric method" for rational and algebraic approximation to algebraic numbers. Consequently, if a; b and n are given positive integers with n 3, we show that the equation of the title possesses at most one solution in positive integers x; y. Further result...

The barycentric interpolation formula defines a stable algorithm for evaluation at points in [−1, 1] of polynomial interpolants through data on Chebyshev grids. Here it is shown that for evaluation at points in the complex plane outside [−1, 1], the algorithm becomes unstable and should be replaced by the alternative modified Lagrange or “first barycentric” formula dating to Jacobi in 1825. Thi...

*Correspondence: [email protected] Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran Abstract In this paper, we solve a time-space fractional diffusion equation. Our methods are based on normalized Bernstein polynomials. For the space domain, we use a set of normalized Bernstein polynomials and for the time domain, which is a semi-infinite domain, ...

The barycentric interpolation formula defines a stable algorithm for evaluation at points in [−1, 1] of polynomial interpolants through data on Chebyshev grids. Here it is shown that for evaluation at points in the complex plane outside [−1, 1], the algorithm becomes unstable and should be replaced by the alternative modified Lagrange or “first barycentric” formula dating to Jacobi in 1825. Thi...

Dawson's Integral is uðyÞ expðÀy 2 Þ R y 0 expðz 2 Þdz. We show that by solving the differential equation du=dy þ 2yu ¼ 1 using the orthogonal rational Chebyshev functions of the second kind, SB 2n ðy; LÞ, which generates a pentadiagonal Petrov–Galerkin matrix, one can obtain an accuracy of roughly ð3=8ÞN digits where N is the number of terms in the spectral series. The SB series is not as effi...

Let f(z) be analytic at the origin, and for e >0, let f(ez) be best approximated in the Chebyshev sense on the unit disk by a rational function of type (m, n). It has been shown previously by the CF method that the error curve for this approximation deviates from a circle by at most O(e 2m+2n+3) as e 0. We prove here that this bound is sharp in two senses: the error curve for a given function c...

Tthis paper, is concerned with obtaining numerical solutions for a class of convection-diffusion equations (CDEs) with variable coefficients. Our approaches are based on collocation methods. These approaches implementing all four kinds of shifted Chebyshev polynomials in combination with Sinc functions to introduce an approximate solution for CDEs . This approximate solution can be expressed as...

in this paper, some results of the chebyshev type integral inequality for the pseudo-integral are proven. the obtained results, are related to the measure of a level set of the maximum and the sum of two non-negative integrable functions. finally, we applied our results  to the case of comonotone functions.