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Fungrim entry: 21241f

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

n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
RisingFactorial(z)k\left(z\right)_{k} Rising factorial
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
StirlingCycle[nk]\left[{n \atop k}\right] Unsigned Stirling number of the first kind
Powab{a}^{b} Power
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("21241f"),
    Formula(Equal(RisingFactorial(x, n), Sum(Mul(StirlingCycle(n, k), Pow(x, k)), Tuple(k, 0, n)))),
    Variables(x, n),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), 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.

2019-08-21 11:44:15.926409 UTC