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

Chebyshev polynomials