This article considers the extension of V. A. Markov’s theorem for polynomial derivatives to polynomials with unit bound on the closed unit ball of any real normed linear space. We show that this extension is equivalent to an inequality for certain directional derivatives of polynomials in two variables that have unit bound on the Chebyshev nodes. We obtain a sharpening of the Markov inequality...

The primary goal of this paper is the study of polynomials with integer coefficients that minimize the sup norm on the set E. In particular, we consider the asymptotic behavior of these polynomials and of their zeros. Let Pn(C) and Pn(Z) be the classes of algebraic polynomials of degree at most n, respectively with complex and with integer coefficients. The problem of minimizing the uniform nor...

We consider the evaluation of a recent generalization of the Epstein-Hubbell elliptic-type integral using the tau method approximation with a Chebyshev polynomial basis. This also leads to an approximation of Lauricella’s hypergeometric function of three variables. Numerical results are given for polynomial approximations of degree 6.

In this paper we provide an extension of the Chebyshev orthogonal rational functions with arbitrary real poles outside [−1, 1] to arbitrary complex poles outside [−1, 1]. The zeros of these orthogonal rational functions are not necessarily real anymore. By using the related para-orthogonal functions, however, we obtain an expression for the nodes and weights for rational Gauss-Chebyshev quadrat...

In this note, we consider a question of Móri regarding estimating the deviation of the kth terms of two discrete probability distributions in terms of the supremum distance between their generating functions over the interval [0, 1]. An optimal bound for distributions on finite support is obtained. Properties of Chebyshev polynomials are employed.

a0Tn(x) + a1Tn−1(x) + · · ·+ amTn−m(x) where (a0, a1, . . . , am) is a fixed m-tuple of real numbers, a0, am 6= 0, Ti(x) are Chebyshev polynomials of the first kind, n = m, m + 1, m + 2, . . . Here we analyze the structure of the set of zeros of such polynomial, depending on A and its limit points when n tends to infinity. Also the expression of envelope of the polynomial is given. An applicati...

Several authors have examined connections between permutations which avoid 132, continued fractions, and Chebyshev polynomials of the second kind. In this paper we prove analogues of some of these results for permutations which avoid 1243 and 2143. Using tools developed to prove these analogues, we give enumerations and generating functions for permutations which avoid 1243, 2143, and certain a...

Gottlieb polynomials were introduced and investigated in 1938, and then have been cited in several articles. Very recently Khan and Akhlaq introduced and investigated Gottlieb polynomials in two and three variables to give their generating functions. Subsequently, Khan and Asif investigated the generating functions for the q-analogue of Gottlieb polynomials. In this sequel, by modifying Khan an...

Given a probability measure μ with infinite support on the unit circle ∂D = {z : |z| = 1}, we consider a sequence of paraorthogonal polynomials hn(z, λ) vanishing at z = λ where λ ∈ ∂D is fixed. We prove that for any fixed z0 6∈ supp(dμ) distinct from λ, we can find an explicit ρ > 0 independent of n such that either hn or hn+1 (or both) has no zero inside the disk B(z0, ρ), with the possible e...