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# 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

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