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Fungrim entry: 4f3e30

Tn ⁣(x)=n2k=0n/2(1)k(nk1)!k!(n2k)!(2x)n2kT_{n}\!\left(x\right) = \frac{n}{2} \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} \frac{{\left(-1\right)}^{k} \left(n - k - 1\right)!}{k ! \left(n - 2 k\right)!} {\left(2 x\right)}^{n - 2 k}
Assumptions:nZ1andxCn \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C}
TeX:
T_{n}\!\left(x\right) = \frac{n}{2} \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} \frac{{\left(-1\right)}^{k} \left(n - k - 1\right)!}{k ! \left(n - 2 k\right)!} {\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
ChebyshevTTn ⁣(x)T_{n}\!\left(x\right) Chebyshev polynomial of the first kind
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
Powab{a}^{b} Power
Factorialn!n ! Factorial
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("4f3e30"),
    Formula(Equal(ChebyshevT(n, x), Mul(Div(n, 2), Sum(Mul(Div(Mul(Pow(-1, k), Factorial(Sub(Sub(n, k), 1))), Mul(Factorial(k), Factorial(Sub(n, Mul(2, k))))), Pow(Mul(2, x), Sub(n, Mul(2, k)))), Tuple(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.

2019-08-17 11:32:46.829430 UTC