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

Legendre polynomials