# 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

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