Fungrim entry: 35e13b

U'_{n}(x) = \frac{\left(n + 1\right) T_{n + 1}\!\left(x\right) - x U_{n}\!\left(x\right)}{{x}^{2} - 1}

Assumptions:

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

TeX:

U'_{n}(x) = \frac{\left(n + 1\right) T_{n + 1}\!\left(x\right) - x U_{n}\!\left(x\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
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second kind
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`Pow`	${a}^{b}$	Power
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("35e13b"),
    Formula(Equal(ComplexDerivative(ChebyshevU(n, x), For(x, x)), Div(Sub(Mul(Add(n, 1), ChebyshevT(Add(n, 1), x)), Mul(x, ChebyshevU(n, 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