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Fungrim entry: 9d66de

Un(r)(x)=(n+1)r+1(nr+1)r(2r+1)!!2F1 ⁣(r+n+2,rn,32+r,1x2){U}^{(r)}_{n}(x) = \frac{\left(n + 1\right)_{r + 1} \left(n - r + 1\right)_{r}}{\left(2 r + 1\right)!!} \,{}_2F_1\!\left(r + n + 2, r - n, \frac{3}{2} + r, \frac{1 - x}{2}\right)
Assumptions:nZandrZ0andxCand(rnorx1)n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, r \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(r \le n \,\mathbin{\operatorname{or}}\, x \ne -1\right)
TeX:
{U}^{(r)}_{n}(x) = \frac{\left(n + 1\right)_{r + 1} \left(n - r + 1\right)_{r}}{\left(2 r + 1\right)!!} \,{}_2F_1\!\left(r + n + 2, r - n, \frac{3}{2} + 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} \,\mathbin{\operatorname{and}}\, \left(r \le n \,\mathbin{\operatorname{or}}\, x \ne -1\right)
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
Hypergeometric2F12F1 ⁣(a,b,c,z)\,{}_2F_1\!\left(a, b, c, z\right) Gauss hypergeometric function
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:
Entry(ID("9d66de"),
    Formula(Equal(ComplexDerivative(ChebyshevU(n, x), x, x, r), Mul(Div(Mul(RisingFactorial(Add(n, 1), Add(r, 1)), RisingFactorial(Add(Sub(n, r), 1), r)), DoubleFactorial(Add(Mul(2, r), 1))), Hypergeometric2F1(Add(Add(r, n), 2), Sub(r, n), Add(Div(3, 2), r), Div(Sub(1, x), 2))))),
    Variables(n, r, x),
    Assumptions(And(Element(n, ZZ), Element(r, ZZGreaterEqual(0)), Element(x, CC), Or(LessEqual(r, n), Unequal(x, -1)))))

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

2019-08-19 14:38:23.809000 UTC