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Fungrim entry: e1797b

Un(r)(x)=π(n+1)2(x1)r3F2 ⁣(1,n,n+2,32,1r,1x2){U}^{(r)}_{n}(x) = \frac{\sqrt{\pi} \left(n + 1\right)}{2 {\left(x - 1\right)}^{r}} \,{}_3{\textbf F}_2\!\left(1, -n, n + 2, \frac{3}{2}, 1 - r, \frac{1 - x}{2}\right)
Assumptions:nZ  and  rZ0  and  xC{1,1}n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \setminus \left\{-1, 1\right\}
{U}^{(r)}_{n}(x) = \frac{\sqrt{\pi} \left(n + 1\right)}{2 {\left(x - 1\right)}^{r}} \,{}_3{\textbf F}_2\!\left(1, -n, n + 2, \frac{3}{2}, 1 - r, \frac{1 - x}{2}\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \setminus \left\{-1, 1\right\}
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
Sqrtz\sqrt{z} Principal square root
Piπ\pi The constant pi (3.14...)
Powab{a}^{b} Power
ZZZ\mathbb{Z} Integers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
Source code for this entry:
    Formula(Equal(ComplexDerivative(ChebyshevU(n, x), For(x, x, r)), Mul(Div(Mul(Sqrt(Pi), Add(n, 1)), Mul(2, Pow(Sub(x, 1), r))), Hypergeometric3F2Regularized(1, Neg(n), Add(n, 2), Div(3, 2), Sub(1, r), Div(Sub(1, x), 2))))),
    Variables(n, r, x),
    Assumptions(And(Element(n, ZZ), Element(r, ZZGreaterEqual(0)), 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.

2021-03-15 19:12:00.328586 UTC