# Fungrim entry: a9a610

$\frac{{\left({e}^{x} - 1\right)}^{k}}{k !} = \sum_{n=k}^{\infty} \left\{{n \atop k}\right\} \frac{{x}^{n}}{n !}$
Assumptions:$k \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}$
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
Pow${a}^{b}$ Power
Exp${e}^{z}$ Exponential function
Factorial$n !$ Factorial
Sum$\sum_{n} f(n)$ Sum
StirlingS2$\left\{{n \atop k}\right\}$ Stirling number of the second kind
Infinity$\infty$ Positive infinity
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
CC$\mathbb{C}$ 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))))

## Topics using this entry

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

2021-03-15 19:12:00.328586 UTC