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ID: INPUT{id.yaml}

Title: Generalized Bernoulli numbers

Definition: >
  Let $\chi$ be a Dirichlet character of modulus $q$.
  The generalized Bernoulli numbers $B_{k,\chi}$ attached to $\chi$
  are defined as the Taylor coefficients of 
  $\sum_{a=1}^q \chi(a) \frac{te^{at}}{e^{qt}-1} = \sum_{k=0}^\infty B_{k,\chi}\frac{t^k}{k!}$.

Parameters:
  q:
    title: modulus of $\chi$
    type: Z
    constraints: $q \geq 1$
  n:
    title: HREF{#CL}[Conrey index] of $\chi$
    type: Z
    constraints: $1 \leq n \leq \max(q-1,1)$
    comments: >
      The pair $(q,n)$ uniquely determines the Dirichlet character $\chi$.
  k:
    type: Z
    constraints: $k \geq 0$

Comments: 
  comment-trivial-zeros: >
    If $\chi(-1) \neq (-1)^k$, we have $B_{k,\chi}=0$ unless $(q,n)=(1,1)$ (that is, $\chi=1$).
    Thus we skip these trivial zero entries.

Formulas:

Programs: 
  program-sage:
    language: Sage
    code: |
      chi = DirichletGroup(3).an_element()
      numbers = [chi.bernoulli(n) for n in [0..10]]

References:

Links:
  CL:
    title: "LMFDB knowl: Conrey label"
    url: https://www.lmfdb.org/knowledge/show/character.dirichlet.conrey
  Wiki:
    title: "Wikipedia: Generalized Bernoulli numbers"
    url: https://en.wikipedia.org/wiki/Bernoulli_number#Generalized_Bernoulli_numbers

Similar tables:
  
Keywords:
  
Tags:
- sequence
- generating function
- special values

Data properties:
  type: C
  
Display properties:
  number-header: $B_{k,\chi}$

Numbers: INPUT{numbers.yaml}

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Generalized Bernoulli numbers

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Numbers

$q$

$n$

$k$

$B_{k,\chi}$

INPUT{numbers.yaml} (not shown in preview)

Definition

Let $\chi$ be a Dirichlet character of modulus $q$. The generalized Bernoulli numbers $B_{k,\chi}$ attached to $\chi$ are defined as the Taylor coefficients of $\sum_{a=1}^q \chi(a) \frac{te^{at}}{e^{qt}-1} = \sum_{k=0}^\infty B_{k,\chi}\frac{t^k}{k!}$.

Parameters

$q$

— modulus of $\chi$ ($q \geq 1$)

$n$

— Conrey index of $\chi$ ($1 \leq n \leq \max(q-1,1)$)

$k$

— integer ($k \geq 0$)

Comments

(1)

If $\chi(-1) \neq (-1)^k$, we have $B_{k,\chi}=0$ unless $(q,n)=(1,1)$ (that is, $\chi=1$). Thus we skip these trivial zero entries.

Programs

(P1)

Sage

chi = DirichletGroup(3).an_element()
numbers = [chi.bernoulli(n) for n in [0..10]]

Links

[1]

LMFDB knowl: Conrey label

[2]

Wikipedia: Generalized Bernoulli numbers

Data properties

Entries are of type: complex number