Minimal polynomials of some beta-numbers and Chebyshev polynomials
نویسنده
چکیده
We consider the β-expansion of 1 which encodes a rational rotation on R/Z under a certain partition. Via transforming its βpolynomial into a linear combination of Chebyshev polynomials, we specify the minimal polynomial of β over the field of rationals. At the end, some number theoretical conjectures are posed with computational evidence to support them.
منابع مشابه
Modified degenerate Carlitz's $q$-bernoulli polynomials and numbers with weight ($alpha ,beta $)
The main goal of the present paper is to construct some families of the Carlitz's $q$-Bernoulli polynomials and numbers. We firstly introduce the modified Carlitz's $q$-Bernoulli polynomials and numbers with weight ($_{p}$. We then define the modified degenerate Carlitz's $q$-Bernoulli polynomials and numbers with weight ($alpha ,beta $) and obtain some recurrence relations and other identities...
متن کاملA numerical Algorithm Based on Chebyshev Polynomials for Solving some Inverse Source Problems
In this paper, two inverse problems of determining an unknown source term in a parabolic equation are considered. First, the unknown source term is estimated in the form of a combination of Chebyshev functions. Then, a numerical algorithm based on Chebyshev polynomials is presented for obtaining the solution of the problem. For solving the problem, the operational matrices of int...
متن کاملA fractional type of the Chebyshev polynomials for approximation of solution of linear fractional differential equations
In this paper we introduce a type of fractional-order polynomials based on the classical Chebyshev polynomials of the second kind (FCSs). Also we construct the operational matrix 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.
متن کاملThe Tangent Analogues of the Chebyshev Polynomials
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 ...
متن کاملA spectral method based on the second kind Chebyshev polynomials for solving a class of fractional optimal control problems
In this paper, we consider the second-kind Chebyshev polynomials (SKCPs) for the numerical solution of the fractional optimal control problems (FOCPs). Firstly, an introduction of the fractional calculus and properties of the shifted SKCPs are given and then operational matrix of fractional integration is introduced. Next, these properties are used together with the Legendre-Gauss quadrature fo...
متن کامل