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

Bernoulli numbers and polynomials