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Fungrim entry: b01280

(log ⁣(1+x))kk!=n=k(1)nk[nk]xnn!\frac{{\left(\log\!\left(1 + x\right)\right)}^{k}}{k !} = \sum_{n=k}^{\infty} {\left(-1\right)}^{n - k} \left[{n \atop k}\right] \frac{{x}^{n}}{n !}
Assumptions:kZ0andxCandx<1k \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|x\right| \lt 1
TeX:
\frac{{\left(\log\!\left(1 + x\right)\right)}^{k}}{k !} = \sum_{n=k}^{\infty} {\left(-1\right)}^{n - k} \left[{n \atop k}\right] \frac{{x}^{n}}{n !}

k \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|x\right| \lt 1
Definitions:
Fungrim symbol Notation Short description
Powab{a}^{b} Power
Loglog ⁣(z)\log\!\left(z\right) Natural logarithm
Factorialn!n ! Factorial
StirlingCycle[nk]\left[{n \atop k}\right] Unsigned Stirling number of the first kind
Infinity\infty Positive infinity
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
Absz\left|z\right| Absolute value
Source code for this entry:
Entry(ID("b01280"),
    Formula(Equal(Div(Pow(Log(Add(1, x)), k), Factorial(k)), Sum(Mul(Mul(Pow(-1, Sub(n, k)), StirlingCycle(n, k)), Div(Pow(x, n), Factorial(n))), Tuple(n, k, Infinity)))),
    Variables(x, k),
    Assumptions(And(Element(k, ZZGreaterEqual(0)), Element(x, CC), Less(Abs(x), 1))))

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2019-06-18 07:49:59.356594 UTC