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Fungrim entry: b6b014

Un(r)(1)=(n+1)r+1(nr+1)r(2r+1)!!{U}^{(r)}_{n}(1) = \frac{\left(n + 1\right)_{r + 1} \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:
{U}^{(r)}_{n}(1) = \frac{\left(n + 1\right)_{r + 1} \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
ChebyshevUUn ⁣(x)U_{n}\!\left(x\right) Chebyshev polynomial of the second 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("b6b014"),
    Formula(Equal(ComplexDerivative(ChebyshevU(n, x), x, 1, r), Div(Mul(RisingFactorial(Add(n, 1), Add(r, 1)), RisingFactorial(Add(Sub(n, r), 1), r)), DoubleFactorial(Add(Mul(2, r), 1))))),
    Variables(n, r),
    Assumptions(And(Element(n, ZZ), Element(r, ZZGreaterEqual(0)))))

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2019-08-17 11:32:46.829430 UTC