Fungrim entry: 59e5df

\left|\frac{d^{2}}{{d x}^{2}} P_{n}\!\left(x\right)\right| \le \frac{\left(n - 1\right) n \left(n + 1\right) \left(n + 2\right)}{8}

Assumptions:

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, -1 \le x \le 1

TeX:

\left|\frac{d^{2}}{{d x}^{2}} P_{n}\!\left(x\right)\right| \le \frac{\left(n - 1\right) n \left(n + 1\right) \left(n + 2\right)}{8}

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, -1 \le x \le 1

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Derivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Derivative
`LegendrePolynomial`	$P_{n}\!\left(z\right)$	Legendre polynomial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("59e5df"),
    Formula(LessEqual(Abs(Derivative(LegendrePolynomial(n, x), Tuple(x, x, 2))), Div(Mul(Mul(Mul(Sub(n, 1), n), Add(n, 1)), Add(n, 2)), 8))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), LessEqual(-1, x, 1))))

Topics using this entry

Legendre polynomials