# Fungrim entry: 3e0817

$B_{n,\chi} = {q}^{n - 1} \sum_{a=1}^{q} \chi(a) B_{n}\!\left(\frac{a}{q}\right)$
Assumptions:$q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}$
TeX:
B_{n,\chi} = {q}^{n - 1} \sum_{a=1}^{q} \chi(a) B_{n}\!\left(\frac{a}{q}\right)

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
GeneralizedBernoulliB$B_{n,\chi}$ Generalized Bernoulli number
Pow${a}^{b}$ Power
Sum$\sum_{n} f(n)$ Sum
BernoulliPolynomial$B_{n}\!\left(z\right)$ Bernoulli polynomial
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
DirichletGroup$G_{q}$ Dirichlet characters with given modulus
Source code for this entry:
Entry(ID("3e0817"),
Formula(Equal(GeneralizedBernoulliB(n, chi), Mul(Pow(q, Sub(n, 1)), Sum(Mul(chi(a), BernoulliPolynomial(n, Div(a, q))), For(a, 1, q))))),
Variables(q, chi, n),
Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), Element(n, 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