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Fungrim entry: e44796

Bn,χ=a=1qχ(a)k=0n(nk)Bkankqk1B_{n,\chi} = \sum_{a=1}^{q} \chi(a) \sum_{k=0}^{n} {n \choose k} B_{k} {a}^{n - k} {q}^{k - 1}
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} = \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
GeneralizedBernoulliBBn,χB_{n,\chi} Generalized Bernoulli number
Sumnf(n)\sum_{n} f(n) Sum
Binomial(nk){n \choose k} Binomial coefficient
BernoulliBBnB_{n} Bernoulli number
Powab{a}^{b} Power
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("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)))))

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

2019-11-11 15:50:15.016492 UTC