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Fungrim entry: 6582c4

Tn(r)(x)=π(x1)r3F2 ⁣(1,n,n,12,1r,1x2){T}^{(r)}_{n}(x) = \frac{\sqrt{\pi}}{{\left(x - 1\right)}^{r}} \,{}_3{\textbf F}_2\!\left(1, -n, n, \frac{1}{2}, 1 - r, \frac{1 - x}{2}\right)
Assumptions:nZandrZ0andxC{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\}
References:
  • http://functions.wolfram.com/Polynomials/ChebyshevT/20/02/01/0002/
TeX:
{T}^{(r)}_{n}(x) = \frac{\sqrt{\pi}}{{\left(x - 1\right)}^{r}} \,{}_3{\textbf F}_2\!\left(1, -n, n, \frac{1}{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\}
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
Sqrtz\sqrt{z} Principal square root
ConstPiπ\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:
Entry(ID("6582c4"),
    Formula(Equal(ComplexDerivative(ChebyshevT(n, x), For(x, x, r)), Mul(Div(Sqrt(ConstPi), Pow(Sub(x, 1), r)), Hypergeometric3F2Regularized(1, Neg(n), n, Div(1, 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))))),
    References("http://functions.wolfram.com/Polynomials/ChebyshevT/20/02/01/0002/"))

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

2019-10-05 13:11:19.856591 UTC