# BernoulliB

BernoulliB[n]

gives the Bernoulli number .

BernoulliB[n,x]

gives the Bernoulli polynomial .

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• The Bernoulli polynomials satisfy the generating function relation .
• The Bernoulli numbers are given by .
• For odd , the Bernoulli numbers are equal to 0, except .
• BernoulliB can be evaluated to arbitrary numerical precision.
• BernoulliB automatically threads over lists.

# Examples

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## Basic Examples(2)

First 10 Bernoulli numbers:

Bernoulli polynomials:

## Scope(3)

Plot Bernoulli polynomials:

## Applications(6)

Find sums of powers using BernoulliB (Faulhaber's formula):

Compare with direct summation:

Set up an EulerMaclaurin integration formula:

Use it for :

Compare with the exact summation result:

Plot roots of Bernoulli polynomials in the complex plane:

Show the approach of Bernoulli numbers to a limiting form:

The denominator of Bernoulli numbers is given by the von StaudtClausen formula:

Compute Bernoulli numbers in modular arithmetic modulo a prime:

## Properties & Relations(3)

Find BernoulliB numbers from their generating function:

Find BernoulliB polynomials from their generating function:

BernoulliB can be represented as a DifferenceRoot:

## Possible Issues(2)

Algorithmically produced results are frequently expressed using Zeta instead of BernoulliB:

When entered in the traditional form, is not automatically interpreted as a Bernoulli number:

## Neat Examples(3)

Going from Bernoulli numbers to Bernoulli polynomials with umbral calculus:

The 20000 Bernoulli number can be computed in under a second:

Define a Hankel matrix whose entries are the Bernoulli numbers:

Its determinant can be expressed in terms of the Barnes G-function:

Wolfram Research (1988), BernoulliB, Wolfram Language function, https://reference.wolfram.com/language/ref/BernoulliB.html (updated 2008).

#### Text

Wolfram Research (1988), BernoulliB, Wolfram Language function, https://reference.wolfram.com/language/ref/BernoulliB.html (updated 2008).

#### CMS

Wolfram Language. 1988. "BernoulliB." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2008. https://reference.wolfram.com/language/ref/BernoulliB.html.

#### APA

Wolfram Language. (1988). BernoulliB. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BernoulliB.html

#### BibTeX

@misc{reference.wolfram_2024_bernoullib, author="Wolfram Research", title="{BernoulliB}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/BernoulliB.html}", note=[Accessed: 12-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_bernoullib, organization={Wolfram Research}, title={BernoulliB}, year={2008}, url={https://reference.wolfram.com/language/ref/BernoulliB.html}, note=[Accessed: 12-September-2024 ]}