Fungrim entry: 9395fc

P_{n}\!\left(z\right) = \,{}_2F_1\!\left(-n, n + 1, 1, \frac{1 - z}{2}\right)

Assumptions:

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

TeX:

P_{n}\!\left(z\right) = \,{}_2F_1\!\left(-n, n + 1, 1, \frac{1 - z}{2}\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`Hypergeometric2F1`	$\,{}_2F_1\!\left(a, b, c, z\right)$	Gauss hypergeometric function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("9395fc"),
    Formula(Equal(LegendrePolynomial(n, z), Hypergeometric2F1(Neg(n), Add(n, 1), 1, Div(Sub(1, z), 2)))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, CC))))

Topics using this entry

Legendre polynomials

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC