Fungrim entry: 4f3e30

T_{n}\!\left(x\right) = \frac{n}{2} \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} \frac{{\left(-1\right)}^{k} \left(n - k - 1\right)!}{k ! \left(n - 2 k\right)!} {\left(2 x\right)}^{n - 2 k}

Assumptions:

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

TeX:

T_{n}\!\left(x\right) = \frac{n}{2} \sum_{k=0}^{\left\lfloor n / 2 \right\rfloor} \frac{{\left(-1\right)}^{k} \left(n - k - 1\right)!}{k ! \left(n - 2 k\right)!} {\left(2 x\right)}^{n - 2 k}

n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevT`	$T_{n}\!\left(x\right)$	Chebyshev polynomial of the first kind
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("4f3e30"),
    Formula(Equal(ChebyshevT(n, x), Mul(Div(n, 2), Sum(Mul(Div(Mul(Pow(-1, k), Factorial(Sub(Sub(n, k), 1))), Mul(Factorial(k), Factorial(Sub(n, Mul(2, k))))), Pow(Mul(2, x), Sub(n, Mul(2, k)))), For(k, 0, Floor(Div(n, 2))))))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(x, CC))))

Topics using this entry

Chebyshev polynomials