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Fungrim entry: a68f0e

Tn(r)(1)=(n)r(nr+1)r(2r1)!!{T}^{(r)}_{n}(1) = \frac{\left(n\right)_{r} \left(n - r + 1\right)_{r}}{\left(2 r - 1\right)!!}
Assumptions:nZandrZ0n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0}
TeX:
{T}^{(r)}_{n}(1) = \frac{\left(n\right)_{r} \left(n - r + 1\right)_{r}}{\left(2 r - 1\right)!!}

n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0}
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
RisingFactorial(z)k\left(z\right)_{k} Rising factorial
ZZZ\mathbb{Z} Integers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("a68f0e"),
    Formula(Equal(ComplexDerivative(ChebyshevT(n, x), x, 1, r), Div(Mul(RisingFactorial(n, r), RisingFactorial(Add(Sub(n, r), 1), r)), DoubleFactorial(Sub(Mul(2, r), 1))))),
    Variables(n, r),
    Assumptions(And(Element(n, ZZ), Element(r, ZZGreaterEqual(0)))))

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

2019-08-19 14:38:23.809000 UTC