Fungrim entry: f5fa23

U_{n}\!\left(x\right) = {2}^{n} \prod_{k=1}^{n} \left(x - \cos\!\left(\frac{k}{n + 1} \pi\right)\right)

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

U_{n}\!\left(x\right) = {2}^{n} \prod_{k=1}^{n} \left(x - \cos\!\left(\frac{k}{n + 1} \pi\right)\right)

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second 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("f5fa23"),
    Formula(Equal(ChebyshevU(n, x), Mul(Pow(2, n), Product(Parentheses(Sub(x, Cos(Mul(Div(k, Add(n, 1)), Pi)))), For(k, 1, n))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(x, CC))))

Topics using this entry

Chebyshev polynomials