Fungrim entry: 50cb6b

U_{n}\!\left(x\right) = \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} {\left(-1\right)}^{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:

U_{n}\!\left(x\right) = \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} {\left(-1\right)}^{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
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second 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("50cb6b"),
    Formula(Equal(ChebyshevU(n, x), Sum(Mul(Mul(Pow(-1, 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

Chebyshev polynomials