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

Dirichlet L-functions