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Fungrim entry: f46e0e

(x−n+1)n=∑k=0ns ⁣(n,k)xk\left(x - n + 1\right)_{n} = \sum_{k=0}^{n} s\!\left(n, k\right) {x}^{k}
Assumptions:n∈Z≥0 and⁡ x∈Cn \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C}
TeX:
\left(x - n + 1\right)_{n} = \sum_{k=0}^{n} s\!\left(n, 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
StirlingS1s ⁣(n,k)s\!\left(n, k\right) Signed Stirling number of the first kind
Powab{a}^{b} Power
ZZGreaterEqualZ≥n\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("f46e0e"),
    Formula(Equal(RisingFactorial(Add(Sub(x, n), 1), n), Sum(Mul(StirlingS1(n, k), Pow(x, k)), 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.

2019-06-18 07:49:59.356594 UTC