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

Stirling numbers