In some applications one is led to consider perturbations of orthogonal polynomials translated by a modification on the first coefficients of the second order recurrence relation satisfied by these polynomials. Moreover, the four Chebyshev families are among the most useful orthogonal sequences due to their exceptional features. Thus, it is important to clarify and explicit the properties of pe...

In this paper we consider second order recurrences {Vk} and {Un} . We give second order linear recurrences for the sequences {V±kn} and {U±kn}. Using these recurrence relations, we derive relationships between the determinants of certain matrices and these sequences. Further, as generalizations of the earlier results, we give representations and trigonometric factorizations of these sequences b...

Riordan matrix methods and properties of generating functions are used to prove that the entries of two Catalan-type Riordan arrays are linked to the Chebyshev polynomials of the first kind. The connections are that the rows of the arrays are used to expand the monomials (1/2) (2x) and (1/2) (4x) in terms of certain Chebyshev polynomials of degree n. In addition, we find new integral representa...

This paper reviews the notion of interpolation of a smooth function by means of Chebyshev polynomials, and the well-known associated results of spectral accuracy when the function is analytic. The rate of decay of the error is proportional to ρ−N , where ρ is a bound on the elliptical radius of the ellipse in which the function has a holomorphic extension. An additional theorem is provided to c...

In this short note, we give simple proofs of several results and conjectures formulated by Stolarsky and Tran concerning generating functions of some families of Chebyshev-like polynomials.

8 In his paper " Lagrange interpolation on Chebyshev points of two variables " (J. 9 220–238), Y. Xu proposed a set of Chebyshev like points for polynomial interpolation in the square 10 [−1, 1] 2 , and derived a compact form of the corresponding Lagrange interpolation formula. We inves-11 tigate computational aspects of the Xu polynomial interpolation formula like numerical stability and 12 ef...

We define a class of multivariate Laurent polynomials closely related to Chebyshev polynomials and prove the simple but somewhat surprising (in view of the fact that the signs of the coefficients of the Chebyshev polynomials themselves alternate) result that their coefficients are non-negative. As a corollary we find that Tn(c cos θ) and Un(c cos θ) are positive definite functions. We further s...

We present a unified local convergence analysis for Chebyshev-Halleytype methods in order to approximate a solution of a nonlinear equation in a Banach space setting. Our methods include the Chebyshev; Halley; super-Halley and other high order methods. The convergence ball and error estimates are given for these methods under the same conditions. Numerical examples are also provided in this stu...

By using complex variable methods (steepest descent and residues) to asymptotically evaluate the coefftcient integrals, the numerical analysis of Hermite function series is discussed. There are striking similarities and differences with the author’s earlier work on Chebyshev polynomial methods (J. Comp. Phys. 45 (1982), 45-49) for infinite or semi-infinite domains. Like Chebyshev series, the He...