# Fungrim entry: 7a85b7

$P_{n}\!\left(z\right) = \frac{1}{{2}^{n}} \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} {\left(-1\right)}^{k} {n \choose k} {2 n - 2 k \choose n} {z}^{n - 2 k}$
Assumptions:$n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}$
TeX:
P_{n}\!\left(z\right) = \frac{1}{{2}^{n}} \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} {\left(-1\right)}^{k} {n \choose k} {2 n - 2 k \choose n} {z}^{n - 2 k}

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
Sum$\sum_{n} f(n)$ Sum
Binomial${n \choose k}$ Binomial coefficient
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("7a85b7"),
Formula(Equal(LegendrePolynomial(n, z), Mul(Div(1, Pow(2, n)), Sum(Mul(Mul(Mul(Pow(-1, k), Binomial(n, k)), Binomial(Sub(Mul(2, n), Mul(2, k)), n)), Pow(z, Sub(n, Mul(2, k)))), For(k, 0, Floor(Div(n, 2))))))),
Variables(n, z),
Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(z, CC))))

## Topics using this entry

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

2020-04-08 16:14:44.404316 UTC