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Fungrim entry: 05fe07

Tn(x)=n(nTn ⁣(x)xUn1 ⁣(x))x21T''_{n}(x) = \frac{n \left(n T_{n}\!\left(x\right) - x U_{n - 1}\!\left(x\right)\right)}{{x}^{2} - 1}
Assumptions:nZandxC{1,1}n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C} \setminus \left\{-1, 1\right\}
TeX:
T''_{n}(x) = \frac{n \left(n T_{n}\!\left(x\right) - x U_{n - 1}\!\left(x\right)\right)}{{x}^{2} - 1}

n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C} \setminus \left\{-1, 1\right\}
Definitions:
Fungrim symbol Notation Short description
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
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
Powab{a}^{b} Power
ZZZ\mathbb{Z} Integers
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("05fe07"),
    Formula(Equal(ComplexDerivative(ChebyshevT(n, x), x, x, 2), Div(Mul(n, Sub(Mul(n, ChebyshevT(n, x)), Mul(x, ChebyshevU(Sub(n, 1), x)))), Sub(Pow(x, 2), 1)))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, SetMinus(CC, Set(-1, 1))))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-09-20 18:07:53.062439 UTC