# 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

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