Fungrim entry: 05fe07

T''_{n}(x) = \frac{n \left(n T_{n}\!\left(x\right) - x U_{n - 1}\!\left(x\right)\right)}{{x}^{2} - 1}

Assumptions:

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \setminus \left\{-1, 1\right\}

TeX:

T''_{n}(x) = \frac{n \left(n T_{n}\!\left(x\right) - x U_{n - 1}\!\left(x\right)\right)}{{x}^{2} - 1}

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C} \setminus \left\{-1, 1\right\}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexDerivative`	$\frac{d}{d z}\, f\!\left(z\right)$	Complex derivative
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`Pow`	${a}^{b}$	Power
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("05fe07"),
    Formula(Equal(ComplexDerivative(ChebyshevT(n, x), For(x, x, 2)), Div(Mul(n, Sub(Mul(n, ChebyshevT(n, x)), Mul(x, ChebyshevU(Sub(n, 1), x)))), Sub(Pow(x, 2), 1)))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZ), Element(x, SetMinus(CC, Set(-1, 1))))))

Topics using this entry

Chebyshev polynomials