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Fungrim entry: 4e914f

Un ⁣(x)=k=0n/2(n+12k+1)(x21)kxn2kU_{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:nZ0andxCn \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
ChebyshevUUn ⁣(x)U_{n}\!\left(x\right) Chebyshev polynomial of the second kind
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
Binomial(nk){n \choose k} Binomial coefficient
Powab{a}^{b} Power
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\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)))), Tuple(k, 0, Floor(Div(n, 2)))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), 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.

2019-08-19 14:38:23.809000 UTC