EulerE

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

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageEulerE✓↗

EulerE(n)the $n$th Euler number

SageMatheuler_number(n)SymPyeuler(n)mpmatheulernum(n)

EulerE(n, x)the $n$th Euler polynomial, evaluated at $x$

SymPyeuler(n, x)mpmatheulerpoly(n, 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 #

live ↗

In$\mathrm{EulerE}(10)$Out$-50\,521$

live ↗

In$\mathrm{EulerE}(8)$Out$1\,385$

Scope #

Listable; odd indices vanish

live ↗

In$\mathrm{EulerE}(\bigl\lbrack0, 1, 2, 3, 4\bigr\rbrack)$Out$\bigl\lbrack1, 0, -1, 0, 5\bigr\rbrack$

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

live ↗

In$\mathrm{EulerE}(2, x)$Out$x^2-x$

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

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In$\mathrm{EulerE}(4, x)$Out$x^4-2x^3+x$

Threads over a list; the signs alternate

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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 $E_3 = 0$

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In$\mathrm{EulerE}(3, \frac{1}{2})$Out$0$

$E_0=1$

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In$\mathrm{EulerE}(0)$Out$1$

Known value · Wolfram MathematicalFunctionData, EulerE

Odd-index Euler numbers vanish

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In$\mathrm{EulerE}(1)$Out$0$

Known value · Wolfram MathematicalFunctionData, EulerE

$E_2=-1$

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In$\mathrm{EulerE}(2)$Out$-1$

Known value · Wolfram MathematicalFunctionData, EulerE

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

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In$\mathrm{EulerE}(3)$Out$0$

Known value · Wolfram MathematicalFunctionData, EulerE

$E_4=5$

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In$\mathrm{EulerE}(4)$Out$5$

Known value · Wolfram MathematicalFunctionData, EulerE

The odd zeros continue: $E_5 = 0$

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In$\mathrm{EulerE}(5)$Out$0$

Known value · Wolfram MathematicalFunctionData, EulerE

$E_6=-61$, with alternating signs

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In$\mathrm{EulerE}(6)$Out$-61$

Known value · Wolfram MathematicalFunctionData, EulerE

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

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In$\mathrm{EulerE}(7)$Out$0$

Known value · Wolfram MathematicalFunctionData, EulerE

$E_9 = 0$

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In$\mathrm{EulerE}(9)$Out$0$

Known value · Wolfram MathematicalFunctionData, EulerE

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

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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

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In$\mathrm{EulerE}(2, 0)$Out$0$

Known value · Wolfram MathematicalFunctionData, EulerE:Polynomial

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

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In$\mathrm{EulerE}(1, 1)$Out$\frac{1}{2}$

Known value · Wolfram MathematicalFunctionData, EulerE:Polynomial

$E_2(1)=0$

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In$\mathrm{EulerE}(2, 1)$Out$0$

Known value · Wolfram MathematicalFunctionData, EulerE:Polynomial

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

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In$\mathrm{EulerE}(0, 1)$Out$1$

Known value · Wolfram MathematicalFunctionData, EulerE:Polynomial

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

Known value · Wolfram MathematicalFunctionData, EulerE:Polynomial, FunctionalEquations

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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 agreeSageMath 13 agreeSymPy 23 agreempmath 21 agree
Crosswalk	Wolfram LanguageSageMathSymPy ×2mpmath ×2

Implementation

mappedsympypackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

See also: BernoulliB