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Fungrim entry: 99aa38

Tn ⁣(x)=n2k=0n/2(1)knk(nkk)(2x)n2kT_{n}\!\left(x\right) = \frac{n}{2} \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} \frac{{\left(-1\right)}^{k}}{n - k} {n - k \choose k} {\left(2 x\right)}^{n - 2 k}
Assumptions:nZ1  and  xCn \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}
T_{n}\!\left(x\right) = \frac{n}{2} \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} \frac{{\left(-1\right)}^{k}}{n - 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}
Fungrim symbol Notation Short description
ChebyshevTTn ⁣(x)T_{n}\!\left(x\right) Chebyshev polynomial of the first 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:
    Formula(Equal(ChebyshevT(n, x), Mul(Div(n, 2), Sum(Mul(Mul(Div(Pow(-1, k), Sub(n, 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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2021-03-15 19:12:00.328586 UTC