# Fungrim entry: 305a29

$T_{n}\!\left(x\right) = {2}^{n - 1} \prod_{k=1}^{n} \left(x - \cos\!\left(\frac{2 k - 1}{2 n} \pi\right)\right)$
Assumptions:$n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}$
TeX:
T_{n}\!\left(x\right) = {2}^{n - 1} \prod_{k=1}^{n} \left(x - \cos\!\left(\frac{2 k - 1}{2 n} \pi\right)\right)

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
ChebyshevT$T_{n}\!\left(x\right)$ Chebyshev polynomial of the first kind
Pow${a}^{b}$ Power
Product$\prod_{n} f(n)$ Product
Cos$\cos(z)$ Cosine
Pi$\pi$ The constant pi (3.14...)
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("305a29"),
Formula(Equal(ChebyshevT(n, x), Mul(Pow(2, Sub(n, 1)), Product(Parentheses(Sub(x, Cos(Mul(Div(Sub(Mul(2, k), 1), Mul(2, n)), Pi)))), For(k, 1, n))))),
Variables(n, x),
Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(x, CC))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC