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Fungrim entry: 50cb6b

Un ⁣(x)=k=0n/2(1)k(nkk)(2x)n2kU_{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:nZ1andxCn \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
ChebyshevUUn ⁣(x)U_{n}\!\left(x\right) Chebyshev polynomial of the second kind
Sumnf(n)\sum_{n} f(n) Sum
Powab{a}^{b} Power
Binomial(nk){n \choose k} Binomial coefficient
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\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))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-10-05 13:11:19.856591 UTC