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Fungrim entry: f55f0a

Pn ⁣(z)=(2nn)(z2)n2F1 ⁣(n2,1n2,12n,1z2)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:nZ0andzC{0}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
LegendrePolynomialPn ⁣(z)P_{n}\!\left(z\right) Legendre polynomial
Binomial(nk){n \choose k} Binomial coefficient
Powab{a}^{b} Power
Hypergeometric2F12F1 ⁣(a,b,c,z)\,{}_2F_1\!\left(a, b, c, z\right) Gauss hypergeometric function
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\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))))))

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

2019-08-17 11:32:46.829430 UTC