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Fungrim entry: a9a610

(ex1)kk!=n=k{nk}xnn!\frac{{\left({e}^{x} - 1\right)}^{k}}{k !} = \sum_{n=k}^{\infty} \left\{{n \atop k}\right\} \frac{{x}^{n}}{n !}
Assumptions:kZ0andxCk \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C}
TeX:
\frac{{\left({e}^{x} - 1\right)}^{k}}{k !} = \sum_{n=k}^{\infty} \left\{{n \atop k}\right\} \frac{{x}^{n}}{n !}

k \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
Powab{a}^{b} Power
Expez{e}^{z} Exponential function
Factorialn!n ! Factorial
Sumnf(n)\sum_{n} f(n) Sum
StirlingS2{nk}\left\{{n \atop k}\right\} Stirling number of the second kind
Infinity\infty Positive infinity
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("a9a610"),
    Formula(Equal(Div(Pow(Sub(Exp(x), 1), k), Factorial(k)), Sum(Mul(StirlingS2(n, k), Div(Pow(x, n), Factorial(n))), For(n, k, Infinity)))),
    Variables(x, k),
    Assumptions(And(Element(k, 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.

2019-11-11 15:50:15.016492 UTC