# Fungrim entry: e1797b

${U}^{(r)}_{n}(x) = \frac{\sqrt{\pi} \left(n + 1\right)}{2 {\left(x - 1\right)}^{r}} \,{}_3{\textbf F}_2\!\left(1, -n, n + 2, \frac{3}{2}, 1 - r, \frac{1 - x}{2}\right)$
Assumptions:$n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; r \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \setminus \left\{-1, 1\right\}$
References:
• http://functions.wolfram.com/Polynomials/ChebyshevU/20/02/01/0002/
TeX:
{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\}
Definitions:
Fungrim symbol Notation Short description
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
ChebyshevU$U_{n}\!\left(x\right)$ Chebyshev polynomial of the second kind
Sqrt$\sqrt{z}$ Principal square root
Pi$\pi$ The constant pi (3.14...)
Pow${a}^{b}$ Power
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("e1797b"),
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))))),
References("http://functions.wolfram.com/Polynomials/ChebyshevU/20/02/01/0002/"))

## 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