Fungrim entry: 4e914f

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

U_{n}\!\left(x\right) = \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} {n + 1 \choose 2 k + 1} {\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
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second 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("4e914f"),
    Formula(Equal(ChebyshevU(n, x), Sum(Mul(Mul(Binomial(Add(n, 1), Add(Mul(2, k), 1)), 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

Chebyshev polynomials