Fungrim entry: a9077a

U_{n}\!\left(x\right) = \sum_{k=0}^{n} \frac{{2}^{k} \left(n + k + 1\right)!}{\left(n - k\right)! \left(2 k + 1\right)!} {\left(x - 1\right)}^{k}

Assumptions:

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

TeX:

U_{n}\!\left(x\right) = \sum_{k=0}^{n} \frac{{2}^{k} \left(n + k + 1\right)!}{\left(n - k\right)! \left(2 k + 1\right)!} {\left(x - 1\right)}^{k}

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

Definitions:

Fungrim symbol	Notation	Short description
`ChebyshevU`	$U_{n}\!\left(x\right)$	Chebyshev polynomial of the second 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("a9077a"),
    Formula(Equal(ChebyshevU(n, x), Sum(Mul(Div(Mul(Pow(2, k), Factorial(Add(Add(n, k), 1))), Mul(Factorial(Sub(n, k)), Factorial(Add(Mul(2, k), 1)))), Pow(Sub(x, 1), k)), For(k, 0, n)))),
    Variables(n, x),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(x, CC))))

Topics using this entry

Chebyshev polynomials