# Fungrim entry: b823b0

${x}^{n} = \sum_{k=0}^{n} \left\{{n \atop k}\right\} \left(x - n + 1\right)_{n}$
Assumptions:$n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}$
TeX:
{x}^{n} = \sum_{k=0}^{n} \left\{{n \atop k}\right\} \left(x - n + 1\right)_{n}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
Pow${a}^{b}$ Power
Sum$\sum_{n} f(n)$ Sum
StirlingS2$\left\{{n \atop k}\right\}$ Stirling number of the second kind
RisingFactorial$\left(z\right)_{k}$ Rising factorial
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("b823b0"),
Formula(Equal(Pow(x, n), Sum(Mul(StirlingS2(n, k), RisingFactorial(Add(Sub(x, n), 1), n)), For(k, 0, n)))),
Variables(x, n),
Assumptions(And(Element(n, 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.

2020-08-27 09:56:25.682319 UTC