EulerE

The Euler numbers $E_n$ and Euler polynomials $E_n(x)$ .

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EulerE(n)the

n

th Euler number

EulerE(n, x)the

n

th Euler polynomial, evaluated at

x

Typed signatures

Type	Package	Applies
`(value, expression?) -> expression \| list<number> \| number`	`number-theory`	always

Domain: Number theory

Details

Odd-indexed Euler numbers past $E_0$ are 0; the even ones come from $\sum_{k=0}^{n/2}\binom{n}{2k}E_{2k}=0$ .
$E_n = 2^n E_n(\frac12)$ ties the numbers to the polynomials.
Not to be confused with BernoulliB or EulerGamma.

Examples

Basic #

In

\mathrm{EulerE}(10)

Out

-50\,521

In

\mathrm{EulerE}(8)

Out

1\,385

Scope #

Listable; odd indices vanish

In

\mathrm{EulerE}(\bigl\lbrack0, 1, 2, 3, 4\bigr\rbrack)

Out

\bigl\lbrack1, 0, -1, 0, 5\bigr\rbrack

the Euler polynomial $E_2(x) = x^2 - x$

In

\mathrm{EulerE}(2, x)

Out

x^2-x

$E_4(x) = x^4 - 2x^3 + x$

In

\mathrm{EulerE}(4, x)

Out

x^4-2x^3+x

Threads over a list; the signs alternate

In

\mathrm{EulerE}(\bigl\lbrack2, 4, 6\bigr\rbrack)

Out

\bigl\lbrack-1, 5, -61\bigr\rbrack

Properties #

$E_n = 2^n E_n(\frac12)$ , and $E_3 = 0$

In

\mathrm{EulerE}(3, \frac{1}{2})

Out

0

$E_0=1$

In

\mathrm{EulerE}(0)

Out

1

Known value · Wolfram MathematicalFunctionData, EulerE

Odd-index Euler numbers vanish

In

\mathrm{EulerE}(1)

Out

0

Known value · Wolfram MathematicalFunctionData, EulerE

$E_2=-1$

In

\mathrm{EulerE}(2)

Out

-1

Known value · Wolfram MathematicalFunctionData, EulerE

$E_3 = 0$ , like every odd-index Euler number

In

\mathrm{EulerE}(3)

Out

0

Known value · Wolfram MathematicalFunctionData, EulerE

$E_4=5$

In

\mathrm{EulerE}(4)

Out

5

Known value · Wolfram MathematicalFunctionData, EulerE

The odd zeros continue: $E_5 = 0$

In

\mathrm{EulerE}(5)

Out

0

Known value · Wolfram MathematicalFunctionData, EulerE

$E_6=-61$ , with alternating signs

In

\mathrm{EulerE}(6)

Out

-61

Known value · Wolfram MathematicalFunctionData, EulerE

$E_7 = 0$ : the secant series has only even terms

In

\mathrm{EulerE}(7)

Out

0

Known value · Wolfram MathematicalFunctionData, EulerE

$E_9 = 0$

In

\mathrm{EulerE}(9)

Out

0

Known value · Wolfram MathematicalFunctionData, EulerE

The Euler polynomial $E_1(x)=x-\tfrac12$ at 0

In

\mathrm{EulerE}(1, 0)

Out

-\frac{1}{2}

Known value · Wolfram MathematicalFunctionData, EulerE:Polynomial

The Euler polynomial $E_2(x)=x^2-x$ vanishes at 0

In

\mathrm{EulerE}(2, 0)

Out

0

Known value · Wolfram MathematicalFunctionData, EulerE:Polynomial

$E_1(1)=\tfrac12$ , symmetric to $E_1(0)$

In

\mathrm{EulerE}(1, 1)

Out

\frac{1}{2}

Known value · Wolfram MathematicalFunctionData, EulerE:Polynomial

$E_2(1)=0$

In

\mathrm{EulerE}(2, 1)

Out

0

Known value · Wolfram MathematicalFunctionData, EulerE:Polynomial

$E_0(x)=1$ for every $x$

In

\mathrm{EulerE}(0, 1)

Out

1

Known value · Wolfram MathematicalFunctionData, EulerE:Polynomial

In

\mathrm{EulerE}(2, \frac{1}{2})+\mathrm{EulerE}(2, \frac{3}{2})

Out

\frac{1}{2}

Known value · Wolfram MathematicalFunctionData, EulerE:Polynomial, FunctionalEquations

In

\mathrm{EulerE}(2, 2)+\mathrm{EulerE}(2, 3)

Out

8

Known value · Wolfram MathematicalFunctionData, EulerE:Polynomial, FunctionalEquations

About

Examples	23
Checked against	Wolfram Language 23 agree
Crosswalk	Wolfram Language

See also: BernoulliB