Fungrim entry: 3c87b9

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

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{1\right\}

TeX:

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

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \left\{1\right\}

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`Pow`	${a}^{b}$	Power
`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("3c87b9"),
    Formula(Equal(LegendrePolynomial(n, z), Mul(Pow(Div(Sub(z, 1), 2), n), Hypergeometric2F1(Neg(n), Neg(n), 1, Div(Add(z, 1), Sub(z, 1)))))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, SetMinus(CC, Set(1))))))

Topics using this entry

Legendre polynomials