# Fungrim entry: 61375f

$U_{n - 1}\!\left(x\right) \sqrt{{x}^{2} - 1} = \frac{1}{2} \left({\left(x + \sqrt{{x}^{2} - 1}\right)}^{n} - {\left(x - \sqrt{{x}^{2} - 1}\right)}^{n}\right)$
Assumptions:$n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}$
TeX:
U_{n - 1}\!\left(x\right) \sqrt{{x}^{2} - 1} = \frac{1}{2} \left({\left(x + \sqrt{{x}^{2} - 1}\right)}^{n} - {\left(x - \sqrt{{x}^{2} - 1}\right)}^{n}\right)

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
ChebyshevU$U_{n}\!\left(x\right)$ Chebyshev polynomial of the second kind
Sqrt$\sqrt{z}$ Principal square root
Pow${a}^{b}$ Power
ZZ$\mathbb{Z}$ Integers
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("61375f"),
Formula(Equal(Mul(ChebyshevU(Sub(n, 1), x), Sqrt(Sub(Pow(x, 2), 1))), Mul(Div(1, 2), Sub(Pow(Add(x, Sqrt(Sub(Pow(x, 2), 1))), n), Pow(Sub(x, Sqrt(Sub(Pow(x, 2), 1))), n))))),
Variables(n, x),
Assumptions(And(Element(n, ZZ), Element(x, CC))))

## 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