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

Dirichlet L-functions