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Fungrim entry: fdf80d

Tn ⁣(x)+Un1 ⁣(x)x21=(x+x21)nT_{n}\!\left(x\right) + U_{n - 1}\!\left(x\right) \sqrt{{x}^{2} - 1} = {\left(x + \sqrt{{x}^{2} - 1}\right)}^{n}
Assumptions:nZandxCn \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C}
TeX:
T_{n}\!\left(x\right) + U_{n - 1}\!\left(x\right) \sqrt{{x}^{2} - 1} = {\left(x + \sqrt{{x}^{2} - 1}\right)}^{n}

n \in \mathbb{Z} \,\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
ChebyshevUUn ⁣(x)U_{n}\!\left(x\right) Chebyshev polynomial of the second kind
Sqrtz\sqrt{z} Principal square root
Powab{a}^{b} Power
ZZZ\mathbb{Z} Integers
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("fdf80d"),
    Formula(Equal(Add(ChebyshevT(n, x), Mul(ChebyshevU(Sub(n, 1), x), Sqrt(Sub(Pow(x, 2), 1)))), Pow(Add(x, Sqrt(Sub(Pow(x, 2), 1))), n))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), 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.

2019-09-22 15:43:45.410764 UTC