# Fungrim entry: ae791d

$T_{n}\!\left(x\right) = \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} {n \choose 2 k} {\left({x}^{2} - 1\right)}^{k} {x}^{n - 2 k}$
Assumptions:$n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}$
TeX:
T_{n}\!\left(x\right) = \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} {n \choose 2 k} {\left({x}^{2} - 1\right)}^{k} {x}^{n - 2 k}

n \in \mathbb{Z}_{\ge 0} \;\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
Sum$\sum_{n} f(n)$ Sum
Binomial${n \choose k}$ Binomial coefficient
Pow${a}^{b}$ Power
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("ae791d"),
Formula(Equal(ChebyshevT(n, x), Sum(Mul(Mul(Binomial(n, Mul(2, k)), Pow(Sub(Pow(x, 2), 1), k)), Pow(x, Sub(n, Mul(2, k)))), For(k, 0, Floor(Div(n, 2)))))),
Variables(n, x),
Assumptions(And(Element(n, ZZGreaterEqual(0)), 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