# EulerE

EulerE[n]

gives the Euler number .

EulerE[n,x]

gives the Euler polynomial .

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• The Euler polynomials satisfy the generating function relation .
• The Euler numbers are given by .
• EulerE automatically threads over lists.

# Examples

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## Basic Examples(2)

First 10 EulerE numbers:

Euler polynomials:

## Scope(4)

Plot Euler polynomials:

Simple exact values are generated automatically:

## Applications(2)

Implement the Boole summation formula:

First a sequence of approximations to :

The sequence converges to the exact answer:

Plot roots of Euler polynomials in the complex plane:

## Properties & Relations(5)

Find Euler numbers from their generating function:

Find Euler polynomials from their generating function:

EulerE can be represented as a DifferenceRoot:

FindSequenceFunction can recognize the EulerE sequence:

The exponential generating function for EulerE:

## Possible Issues(1)

Algorithmically produced results are often expressed using Zeta instead of EulerE:

## Neat Examples(4)

Umbral calculus with Euler numbers:

Histogram of digits of 10000 Euler number:

The sequence of Euler numbers modulo a fixed number is periodic:

Define a Hankel matrix whose entries are the Euler numbers:

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

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

#### Text

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

#### CMS

Wolfram Language. 1988. "EulerE." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/EulerE.html.

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_eulere, author="Wolfram Research", title="{EulerE}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/EulerE.html}", note=[Accessed: 06-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_eulere, organization={Wolfram Research}, title={EulerE}, year={1988}, url={https://reference.wolfram.com/language/ref/EulerE.html}, note=[Accessed: 06-August-2024 ]}