# Fungrim entry: 555e10

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

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
BernoulliPolynomial$B_{n}\!\left(z\right)$ Bernoulli polynomial
Sum$\sum_{n} f(n)$ Sum
Binomial${n \choose k}$ Binomial coefficient
BernoulliB$B_{n}$ Bernoulli number
Pow${a}^{b}$ Power
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("555e10"),
Formula(Equal(BernoulliPolynomial(n, x), Sum(Mul(Mul(Binomial(n, k), BernoulliB(Sub(n, k))), Pow(x, k)), For(k, 0, n)))),
Variables(n, x),
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.

2021-03-15 19:12:00.328586 UTC