Fungrim entry: 925fdf

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

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}

TeX:

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

n \in \mathbb{Z}_{\ge 1} \;\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("925fdf"),
    Formula(Equal(Sub(Add(Mul(Sub(1, Pow(z, 2)), ComplexDerivative(LegendrePolynomial(n, z), For(z, z, 1))), Mul(Mul(n, z), LegendrePolynomial(n, z))), Mul(n, LegendrePolynomial(Sub(n, 1), z))), 0)),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(z, CC))))

Topics using this entry

Legendre polynomials