Incomplete q-Chebyshev Polynomials

In this paper, we get the generating functions of q-Chebyshev polynomials using operator. Also considering explicit formulas of q-Chebyshev polynomials, we give new generalizations of q-Chebyshev polynomials called incomplete q-Chebyshev polynomials of the first and second kind. We obtain recurrence relations and several properties of these polynomials. We show that there are connections between incomplete q-Chebyshev polynomials and the some well-known polynomials. Keywords: q-Chebyshev polynomials, q-Fibonacci polynomials, Incomplete polynomials, Fibonacci numbe

Generalized Chebyshev polynomials of the second kind

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 paper with some results concerning integrals of the generalized Chebyshev-II and Bernstein polynomials.Comment: Change the title (Tschebyscheff to Chebyshev), and adding few comments. Adding the Journal reference

Applications of Integer and Semi-Infinite Programming to the Integer Chebyshev Problem

We consider the integer Chebyshev problem, that of minimizing the supremum norm over polynomials with integer coefficients on the interval $[0,1]$. We implement algorithms from semi-infinite programming and a branch and bound algorithm to improve on previous methods for finding integer Chebyshev polynomials of degree $n$. Using our new method, we found 16 new integer Chebyshev polynomials of degrees in the range 147 to 244.Comment: 12 page

Small polynomials with integer coefficients

We study the problem of minimizing the supremum norm, on a segment of the real line or on a compact set in the plane, by polynomials with integer coefficients. The extremal polynomials are naturally called integer Chebyshev polynomials. Their factors, zero distribution and asymptotics are the main subjects of this paper. In particular, we show that the integer Chebyshev polynomials for any infinite subset of the real line must have infinitely many distinct factors, which answers a question of Borwein and Erd\'{e}lyi. Furthermore, it is proved that the accumulation set for their zeros must be of positive capacity in this case. We also find the first nontrivial examples of explicit integer Chebyshev constants for certain classes of lemniscates. Since it is rarely possible to obtain an exact value of integer Chebyshev constant, good estimates are of special importance. Introducing the methods of weighted potential theory, we generalize and improve the Hilbert-Fekete upper bound for integer Chebyshev constant. These methods also give bounds for the multiplicities of factors of integer Chebyshev polynomials, and lower bounds for integer Chebyshev constant. Moreover, all the mentioned bounds can be found numerically, by using various extremal point techniques, such as weighted Leja points algorithm. Applying our results in the classical case of the segment $[0,1]$, we improve the known bounds for the integer Chebyshev constant and the multiplicities of factors of the integer Chebyshev polynomials.Comment: 35 pages, 1 figur

An inequality on Chebyshev polynomials

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

Permutation properties of Dickson and Chebyshev polynomials and connections to number theory

The $k$th Dickson polynomial of the first kind, $D_k(x) \in {\mathbb Z}[x]$, is determined by the formula: $D_k(u+1/u) = u^k + 1/u^k$, where $k \ge 0$ and $u$ is an indeterminate. These polynomials are closely related to Chebyshev polynomials and have been widely studied. Leonard Eugene Dickson proved in 1896 that $D_k(x)$ is a permutation polynomial on ${\mathbb F}_{p^n}$, $p$ prime, if and only if GCD$(k,p^{2n}-1)=1$, and his result easily carries over to Chebyshev polynomials when $p$ is odd. This article continues on this theme, as we find special subsets of ${\mathbb F}_{p^n}$ that are stabilized or permuted by Dickson or Chebyshev polynomials. Our analysis also leads to a factorization formula for Dickson and Chebyshev polynomials and some new results in elementary number theory. For example, we show that if $q$ is an odd prime power, then \prod\left\{ a \in{\mathbb F}_q^\times : \text{a$and$4-a are nonsquares} \right\} = 2.Comment: 29 pages. This is a revised and expanded version of "Permutations properties of Dickson polynomials and connections to number theory" that was presented at the Mathematical Congress of the Americas, MCA2017. The new version contains a section on Chebyshev polynomials and a section giving alternative proofs that were contributed by Richard Stong, Bradley Brock, and John Dillo

Polynomials invertible in k-radicals

A classic result of Ritt describes polynomials invertible in radicals: they are compositions of power polynomials, Chebyshev polynomials and polynomials of degree at most 4. In this paper we prove that a polynomial invertible in radicals and solutions of equations of degree at most k is a composition of power polynomials, Chebyshev polynomials, polynomials of degree at most k and, if k < 15, certain polynomials with exceptional monodromy groups. A description of these exceptional polynomials is given. The proofs rely on classification of monodromy groups of primitive polynomials obtained by M\"{u}ller based on group-theoretical results of Feit and on previous work on primitive polynomials with exceptional monodromy groups by many authors.Comment: 19 pages, 14 figure

Symmetrized Chebyshev Polynomials

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. We further show that a Central Limit Theorem holds for our polynomials.Comment: Enhancement of math.CA/030121

New hypergeometric connection formulae between Fibonacci and Chebyshev polynomials

We establish new connection formulae between Fibonacci polynomials and Chebyshev polynomials of the first and second kinds. These formulae are expressed in terms of certain values of hypergeometric functions of the type 2F1. Consequently, we obtain some new expressions for the celebrated Fibonacci numbers and their derivatives sequences. Moreover, we evaluate some definite integrals involving products of Fibonacci and Chebyshev polynomials

Application of a composition of generating functions for obtaining explicit formulas of polynomials

Using notions of composita and composition of generating functions we obtain explicit formulas for Chebyshev polynomials, Legendre polynomials, Gegenbauer polynomials, Associated Laguerre polynomials, Stirling polynomials, Abel polynomials, Bernoulli Polynomials of the Second Kind, Generalized Bernoulli polynomials, Euler Polynomials, Peters polynomials, Narumi polynomials, Humbert polynomials, Lerch polynomials and Mahler polynomials