# Fungrim entry: e44796

$B_{n,\chi} = \sum_{a=1}^{q} \chi(a) \sum_{k=0}^{n} {n \choose k} B_{k} {a}^{n - k} {q}^{k - 1}$
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} = \sum_{a=1}^{q} \chi(a) \sum_{k=0}^{n} {n \choose k} B_{k} {a}^{n - k} {q}^{k - 1}

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
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
DirichletGroup$G_{q}$ Dirichlet characters with given modulus
Source code for this entry:
Entry(ID("e44796"),
Formula(Equal(GeneralizedBernoulliB(n, chi), Sum(Mul(chi(a), Sum(Mul(Mul(Mul(Binomial(n, k), BernoulliB(k)), Pow(a, Sub(n, k))), Pow(q, Sub(k, 1))), For(k, 0, n))), 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.

2020-08-27 09:56:25.682319 UTC