Fungrim entry: 473c36

\int_{-1}^{1} U_{n}\!\left(x\right) U_{m}\!\left(x\right) \sqrt{1 - {x}^{2}} \, dx = \frac{\pi}{2} \delta_{(n,m)}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, m \in \mathbb{Z}_{\ge 0}

TeX:

\int_{-1}^{1} U_{n}\!\left(x\right) U_{m}\!\left(x\right) \sqrt{1 - {x}^{2}} \, dx = \frac{\pi}{2} \delta_{(n,m)}

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, m \in \mathbb{Z}_{\ge 0}

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
`ConstPi`	$\pi$	The constant pi (3.14...)
`KroneckerDelta`	$\delta_{(x,y)}$	Kronecker delta
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("473c36"),
    Formula(Equal(Integral(Mul(Mul(ChebyshevU(n, x), ChebyshevU(m, x)), Sqrt(Sub(1, Pow(x, 2)))), Tuple(x, -1, 1)), Mul(Div(ConstPi, 2), KroneckerDelta(n, m)))),
    Variables(n, m),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(m, ZZGreaterEqual(0)))))

Topics using this entry

Chebyshev polynomials