# Fungrim entry: fdf80d

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

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
ChebyshevT$T_{n}\!\left(x\right)$ Chebyshev polynomial of the first kind
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("fdf80d"),
Formula(Equal(Add(ChebyshevT(n, x), Mul(ChebyshevU(Sub(n, 1), x), Sqrt(Sub(Pow(x, 2), 1)))), Pow(Add(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