# Fungrim entry: 99aa38

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

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
Sum$\sum_{n} f(n)$ Sum
Pow${a}^{b}$ Power
Binomial${n \choose k}$ Binomial coefficient
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("99aa38"),
Formula(Equal(ChebyshevT(n, x), Mul(Div(n, 2), Sum(Mul(Mul(Div(Pow(-1, k), Sub(n, k)), Binomial(Sub(n, k), k)), Pow(Mul(2, x), Sub(n, Mul(2, k)))), For(k, 0, Floor(Div(n, 2))))))),
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