# Fungrim entry: 12ce84

${T}^{(r)}_{n}(x) = \frac{\left(n\right)_{r} \left(n - r + 1\right)_{r}}{\left(2 r - 1\right)!!} \,{}_2F_1\!\left(r + n, r - n, \frac{1}{2} + r, \frac{1 - x}{2}\right)$
Assumptions:$n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left(r \le n \;\mathbin{\operatorname{or}}\; x \ne -1\right)$
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
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
ChebyshevT$T_{n}\!\left(x\right)$ Chebyshev polynomial of the first kind
RisingFactorial$\left(z\right)_{k}$ Rising factorial
Hypergeometric2F1$\,{}_2F_1\!\left(a, b, c, z\right)$ Gauss hypergeometric function
ZZ$\mathbb{Z}$ Integers
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
CC$\mathbb{C}$ 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), NotEqual(x, -1)))))

## Topics using this entry

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