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Fungrim entry: 0cbe75

Tn ⁣(x)=12((x+x21)n+(xx21)n)T_{n}\!\left(x\right) = \frac{1}{2} \left({\left(x + \sqrt{{x}^{2} - 1}\right)}^{n} + {\left(x - \sqrt{{x}^{2} - 1}\right)}^{n}\right)
Assumptions:nZ  and  xCn \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}
T_{n}\!\left(x\right) = \frac{1}{2} \left({\left(x + \sqrt{{x}^{2} - 1}\right)}^{n} + {\left(x - \sqrt{{x}^{2} - 1}\right)}^{n}\right)

n \in \mathbb{Z} \;\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
Powab{a}^{b} Power
Sqrtz\sqrt{z} Principal square root
ZZZ\mathbb{Z} Integers
CCC\mathbb{C} Complex numbers
Source code for this entry:
    Formula(Equal(ChebyshevT(n, x), Mul(Div(1, 2), Add(Pow(Add(x, Sqrt(Sub(Pow(x, 2), 1))), n), Pow(Sub(x, Sqrt(Sub(Pow(x, 2), 1))), n))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, CC))))

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2021-03-15 19:12:00.328586 UTC