Fungrim entry: 859445

P_{2 n + 1}\!\left(z\right) = \frac{{\left(-1\right)}^{n}}{{4}^{n}} \left(2 n + 1\right) {2 n \choose n} z \,{}_2F_1\!\left(-n, n + \frac{3}{2}, \frac{3}{2}, {z}^{2}\right)

Assumptions:

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

TeX:

P_{2 n + 1}\!\left(z\right) = \frac{{\left(-1\right)}^{n}}{{4}^{n}} \left(2 n + 1\right) {2 n \choose n} z \,{}_2F_1\!\left(-n, n + \frac{3}{2}, \frac{3}{2}, {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
`Pow`	${a}^{b}$	Power
`Binomial`	${n \choose k}$	Binomial coefficient
`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("859445"),
    Formula(Equal(LegendrePolynomial(Add(Mul(2, n), 1), z), Mul(Mul(Mul(Mul(Div(Pow(-1, n), Pow(4, n)), Add(Mul(2, n), 1)), Binomial(Mul(2, n), n)), z), Hypergeometric2F1(Neg(n), Add(n, Div(3, 2)), Div(3, 2), Pow(z, 2))))),
    Variables(n, z),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, SetMinus(CC)))))

Topics using this entry

Legendre polynomials