Fungrim home page

Fungrim entry: 4aab8a

k=1mkn=Bn+1 ⁣(m+1)Bm+1m+1\sum_{k=1}^{m} {k}^{n} = \frac{B_{n + 1}\!\left(m + 1\right) - B_{m + 1}}{m + 1}
Assumptions:nZ0  and  mZ0n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 0}
\sum_{k=1}^{m} {k}^{n} = \frac{B_{n + 1}\!\left(m + 1\right) - B_{m + 1}}{m + 1}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 0}
Fungrim symbol Notation Short description
Sumnf(n)\sum_{n} f(n) Sum
Powab{a}^{b} Power
BernoulliPolynomialBn ⁣(z)B_{n}\!\left(z\right) Bernoulli polynomial
BernoulliBBnB_{n} Bernoulli number
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
    Formula(Equal(Sum(Pow(k, n), For(k, 1, m)), Div(Sub(BernoulliPolynomial(Add(n, 1), Add(m, 1)), BernoulliB(Add(m, 1))), Add(m, 1)))),
    Variables(n, m),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(m, ZZGreaterEqual(0)))))

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