# Fungrim entry: c8d10e

${\left(P_{n}\!\left(x\right)\right)}^{2} - P_{n - 1}\!\left(x\right) P_{n + 1}\!\left(x\right) \ge 0$
Assumptions:$n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \left[-1, 1\right]$
TeX:
{\left(P_{n}\!\left(x\right)\right)}^{2} - P_{n - 1}\!\left(x\right) P_{n + 1}\!\left(x\right) \ge 0

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \left[-1, 1\right]
Definitions:
Fungrim symbol Notation Short description
Pow${a}^{b}$ Power
LegendrePolynomial$P_{n}\!\left(z\right)$ Legendre polynomial
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
ClosedInterval$\left[a, b\right]$ Closed interval
Source code for this entry:
Entry(ID("c8d10e"),
Formula(GreaterEqual(Sub(Pow(Parentheses(LegendrePolynomial(n, x)), 2), Mul(LegendrePolynomial(Sub(n, 1), x), LegendrePolynomial(Add(n, 1), x))), 0)),
Variables(n, x),
Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(x, ClosedInterval(-1, 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