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euler

Euler numbers and polynomials

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

euler(n)

euler(n, x)

Parameters

n

-

non-negative integer

x

-

expression

Description

• 

The function euler computes the nth Euler number, or the nth Euler polynomial in x.

• 

The nth Euler polynomial E(n, x) is defined by the exponential generating function:

2ⅇtxⅇt+1=n=0En,xtnn!

• 

The nth Euler number E(n) is defined by the exponential generating function:

2ⅇt+ⅇt=n=0Entnn!

• 

The nth Euler polynomial E(n, x) and the nth Euler number E(n) are related by the following equation:

En=2nEn,12

Examples

euler6

61

(1)

euler2,5

20

(2)

euler4,x

x42x3+x

(3)

euler4,12

516

(4)

euler2,5x

25x25x

(5)

See Also

bernoulli

initialfunctions

 


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