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Fungrim entry: c8d10e

(Pn ⁣(x))2Pn1 ⁣(x)Pn+1 ⁣(x)0{\left(P_{n}\!\left(x\right)\right)}^{2} - P_{n - 1}\!\left(x\right) P_{n + 1}\!\left(x\right) \ge 0
Assumptions:nZ1andx[1,1]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
Powab{a}^{b} Power
LegendrePolynomialPn ⁣(z)P_{n}\!\left(z\right) Legendre polynomial
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
ClosedInterval[a,b]\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)))))

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

2019-10-05 13:11:19.856591 UTC