# 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

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

2021-03-15 19:12:00.328586 UTC