Fungrim entry: 3d77ab

\int_{-1}^{1} T_{n}\!\left(x\right) {x}^{m} \frac{1}{\sqrt{1 - {x}^{2}}} \, dx = 0

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \{0, 1, \ldots, n - 1\}

TeX:

\int_{-1}^{1} T_{n}\!\left(x\right) {x}^{m} \frac{1}{\sqrt{1 - {x}^{2}}} \, dx = 0

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \{0, 1, \ldots, n - 1\}

Definitions:

Fungrim symbol	Notation	Short description
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`Pow`	${a}^{b}$	Power
`Sqrt`	$\sqrt{z}$	Principal square root
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints

Source code for this entry:

Entry(ID("3d77ab"),
    Formula(Equal(Integral(Mul(Mul(ChebyshevT(n, x), Pow(x, m)), Div(1, Sqrt(Sub(1, Pow(x, 2))))), For(x, -1, 1)), 0)),
    Variables(n, m),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(m, Range(0, Sub(n, 1))))))

Topics using this entry

Chebyshev polynomials