# Fungrim entry: f0569a

$P_{n}\!\left(z\right) = \sum_{k=0}^{n} {n \choose k} {n + k \choose k} {\left(\frac{z - 1}{2}\right)}^{k}$
Assumptions:$n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C}$
TeX:
P_{n}\!\left(z\right) = \sum_{k=0}^{n} {n \choose k} {n + k \choose k} {\left(\frac{z - 1}{2}\right)}^{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
Sum$\sum_{n} f(n)$ Sum
Binomial${n \choose k}$ Binomial coefficient
Pow${a}^{b}$ Power
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("f0569a"),
Formula(Equal(LegendrePolynomial(n, z), Sum(Mul(Mul(Binomial(n, k), Binomial(Add(n, k), k)), Pow(Div(Sub(z, 1), 2), k)), For(k, 0, n)))),
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.

2021-03-15 19:12:00.328586 UTC