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Fungrim entry: b823b0

xn=k=0n{nk}(xn+1)n{x}^{n} = \sum_{k=0}^{n} \left\{{n \atop k}\right\} \left(x - n + 1\right)_{n}
Assumptions:nZ0andxCn \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C}
TeX:
{x}^{n} = \sum_{k=0}^{n} \left\{{n \atop k}\right\} \left(x - n + 1\right)_{n}

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
Powab{a}^{b} Power
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
StirlingS2{nk}\left\{{n \atop k}\right\} Stirling number of the second kind
RisingFactorial(z)k\left(z\right)_{k} Rising factorial
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("b823b0"),
    Formula(Equal(Pow(x, n), Sum(Mul(StirlingS2(n, k), RisingFactorial(Add(Sub(x, n), 1), n)), Tuple(k, 0, n)))),
    Variables(x, n),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(x, CC))))

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

2019-08-21 11:44:15.926409 UTC