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Fungrim entry: 12ce84

Tn(r)(x)=(n)r(nr+1)r(2r1)!!2F1 ⁣(r+n,rn,12+r,1x2){T}^{(r)}_{n}(x) = \frac{\left(n\right)_{r} \left(n - r + 1\right)_{r}}{\left(2 r - 1\right)!!} \,{}_2F_1\!\left(r + n, r - n, \frac{1}{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:
{T}^{(r)}_{n}(x) = \frac{\left(n\right)_{r} \left(n - r + 1\right)_{r}}{\left(2 r - 1\right)!!} \,{}_2F_1\!\left(r + n, r - n, \frac{1}{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
ChebyshevTTn ⁣(x)T_{n}\!\left(x\right) Chebyshev polynomial of the first 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("12ce84"),
    Formula(Equal(ComplexDerivative(ChebyshevT(n, x), For(x, x, r)), Mul(Div(Mul(RisingFactorial(n, r), RisingFactorial(Add(Sub(n, r), 1), r)), DoubleFactorial(Sub(Mul(2, r), 1))), Hypergeometric2F1(Add(r, n), Sub(r, n), Add(Div(1, 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.

2019-11-19 15:10:20.037976 UTC