Euler polynomials
Polynomials of the form
$$ E _ {n} ( x) = \sum _ { k=0}^ { n } \left ( \begin{array}{c} n \\
k
\end{array}
\right )
\frac{E _ k}{2 ^ {k}}
\left ( x -
\frac{1}{2}
\right ) ^ {n-k} ,
$$
where $ E _ {k} $ are the Euler numbers. The Euler polynomials can be computed successively by means of the formula
$$ E _ {n} ( x) + \sum _ { s=0} ^ { n } \left ( \begin{array}{c} n \\
s
\end{array}
\right )
E _ {s} ( x) = 2 x ^ {n} . $$
In particular,
$$ E _ {0} ( x) = 1 ,\ \ E _ {1} ( x) = x - \frac{1}{2}
,\ \
E _ {2} ( x) = x ( x - 1 ) . $$
The Euler polynomials satisfy the difference equation
$$ E _ {n} ( x + 1 ) + E _ {n} ( x) = 2 x ^ {n} $$
and belong to the class of Appell polynomials, that is, they satisfy
$$
\frac{d}{dx}
E _ {n} ( x) = n E _ {n-} 1 ( x) .
$$
The generating function of the Euler polynomials is
$$
\frac{2 e ^ {xt} }{e ^ {t} + 1 }
= \
\sum _ { n=0}^ \infty \frac{E _ {n} ( x) }{n!}
t ^ {n} .
$$
The Euler polynomials admit the Fourier expansion
$$ \tag{* } E _ {n} ( x) = \frac{n!} {\pi ^ {n+ 1 }} \sum _ { k=0} ^ \infty
\frac{\cos [ ( 2 k + 1 ) \pi x + ( n+ 1) \pi / 2 ] }{( 2 k + 1 ) ^ {n+1} }
,
$$
$$ 0 \leq x \leq 1 ,\ n \geq 1 . $$
They satisfy the relations
$$ E _ {n} ( 1 - x ) = ( - 1 ) ^ {n} E _ {n} ( x) , $$
$$ E _ {n} ( mx) = m ^ {n} \sum _ { k=0} ^ { m-1} ( - 1 ) ^ {k} E _ {n} \left ( x + \frac{k}{m}
\right )
$$
if $ m $ is odd,
$$ E _ {n} ( mx) = - \frac{2 m ^ {n} }{n+1} \sum _ { k=0} ^ { m-1} ( - 1 ) ^ {k} B _ {n+1} \left ( x + \frac{k}{m}
\right )
$$
if $ m $ is even. Here $B_{n+1}$ is a Bernoulli polynomial (cf. Bernoulli polynomials). The periodic functions coinciding with the right-hand side of (*) are extremal in the Kolmogorov inequality and in a number of other extremal problems in function theory. Generalized Euler polynomials have also been considered.
References
| [1] | L. Euler, "Opera omnia: series prima: opera mathematica: institutiones calculi differentialis" , Teubner (1980) (Translated from Latin) |
| [2] | N.E. Nörlund, "Volesungen über Differenzenrechnung" , Springer (1924) |
Comments
The Euler polynomials satisfy in addition the identities
$$ E _ {n} ( x+ h) = $$
$$ = \ E _ {n} ( x) + \left ( \begin{array}{c} n \\
1
\end{array}
\right ) h E _ {n-1} ( x) + \dots +
\left ( \begin{array}{c} n \\
n- 1
\end{array}
\right ) h ^ {n-1} E _ {1} ( x) + E _ {0} ( x),
$$
written symbolically as
$$ E _ {n} ( x+ h) = \{ E ( x) + h \} ^ {n} . $$
Here the right-hand side should be read as follows: first expand the right-hand side into sums of expressions $ ( {} _ {i} ^ {n} ) \{ E ( x) \} ^ {i} h ^ {n-i} $ and then replace $ \{ E ( x) \} ^ {i} $ with $ E _ {i} ( x) $.
Using the same symbolic notation one has for every polynomial $ p( x) $,
$$ p ( E ( x) + 1) + p( E( x) ) = 2 p( x) . $$
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