Fungrim entry: f55f0a

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

Assumptions:

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

TeX:

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

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

Definitions:

Fungrim symbol	Notation	Short description
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`Binomial`	${n \choose k}$	Binomial coefficient
`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("f55f0a"),
    Formula(Equal(LegendrePolynomial(n, z), Mul(Mul(Binomial(Mul(2, n), n), Pow(Div(z, 2), n)), Hypergeometric2F1(Neg(Div(n, 2)), Div(Sub(1, n), 2), Sub(Div(1, 2), n), Div(1, Pow(z, 2)))))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, SetMinus(CC, Set(0))))))

Topics using this entry

Legendre polynomials