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Fungrim entry: a9077a

Un ⁣(x)=k=0n2k(n+k+1)!(nk)!(2k+1)!(x1)kU_{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:nZ1  and  xCn \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}
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}
Fungrim symbol Notation Short description
ChebyshevUUn ⁣(x)U_{n}\!\left(x\right) Chebyshev polynomial of the second kind
Sumnf(n)\sum_{n} f(n) Sum
Powab{a}^{b} Power
Factorialn!n ! Factorial
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
Source code for this entry:
    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))))

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2021-03-15 19:12:00.328586 UTC