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Fungrim entry: 3e0817

Bn,χ=qn1a=1qχ ⁣(a)Bn ⁣(aq)B_{n,\chi} = {q}^{n - 1} \sum_{a=1}^{q} \chi\!\left(a\right) B_{n}\!\left(\frac{a}{q}\right)
Assumptions:qZ1andχGqandnZ0q \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\!\left(a\right) 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
GeneralizedBernoulliBBn,χB_{n,\chi} Generalized Bernoulli number
Powab{a}^{b} Power
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
BernoulliPolynomialBn ⁣(z)B_{n}\!\left(z\right) Bernoulli polynomial
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
DirichletGroupGqG_{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))), Tuple(a, 1, q))))),
    Variables(q, chi, n),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), Element(n, ZZGreaterEqual(0)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-08-19 14:38:23.809000 UTC