Fungrim entry: 27688e

\left(1 - {z}^{2}\right) P''_{n}(z) - 2 z P'_{n}(z) + n \left(n + 1\right) P_{n}\!\left(z\right) = 0

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

\left(1 - {z}^{2}\right) P''_{n}(z) - 2 z P'_{n}(z) + n \left(n + 1\right) P_{n}\!\left(z\right) = 0

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("27688e"),
    Formula(Equal(Add(Sub(Mul(Sub(1, Pow(z, 2)), ComplexDerivative(LegendrePolynomial(n, z), For(z, z, 2))), Mul(Mul(2, z), ComplexDerivative(LegendrePolynomial(n, z), For(z, z, 1)))), Mul(Mul(n, Add(n, 1)), LegendrePolynomial(n, z))), 0)),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, CC))))

Topics using this entry

Legendre polynomials