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(latex details)
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$$  
 
$$  
E _ {n} ( x)  =  \sum _ { k= } 0 ^ { n }  \left ( \begin{array}{c}
+
E _ {n} ( x)  =  \sum _ { k=0}^ { n }  \left ( \begin{array}{c}
 
n \\
 
n \\
 
  k  
 
  k  
Line 31: Line 31:
  
 
$$  
 
$$  
E _ {n} ( x) + \sum _ { s= } 0 ^ { n }  \left ( \begin{array}{c}
+
E _ {n} ( x) + \sum _ { s=0} ^ { n }  \left ( \begin{array}{c}
 
n \\
 
n \\
 
  s  
 
  s  
Line 69: Line 69:
 
\frac{2 e  ^ {xt} }{e  ^ {t} + 1 }
 
\frac{2 e  ^ {xt} }{e  ^ {t} + 1 }
 
   = \  
 
   = \  
\sum _ { n= } 0 ^  \infty   
+
\sum _ { n=0}^  \infty   
 
\frac{E _ {n} ( x) }{n!}
 
\frac{E _ {n} ( x) }{n!}
 
  t  ^ {n} .
 
  t  ^ {n} .
Line 77: Line 77:
  
 
$$ \tag{* }
 
$$ \tag{* }
E _ {n} ( x)  =  n! over {\pi  ^ {n+} 1 } \sum _ { k= } 0 ^  \infty   
+
E _ {n} ( x)  =  n! over {\pi  ^ {n+} 1 } \sum _ { k=0} ^  \infty   
  
 
\frac{\cos [ ( 2 k + 1 ) \pi x + ( n+ 1) \pi / 2 ] }{( 2 k + 1 )  ^ {n+} 1 }
 
\frac{\cos [ ( 2 k + 1 ) \pi x + ( n+ 1) \pi / 2 ] }{( 2 k + 1 )  ^ {n+} 1 }
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$$  
 
$$  
E _ {n} ( mx)  =  m  ^ {n} \sum _ { k= } 0 ^ { m- } 1 ( - 1
+
E _ {n} ( mx)  =  m  ^ {n} \sum _ { k=0} ^ { m-1} ( - 1
 
)  ^ {k} E _ {n} \left ( x +  
 
)  ^ {k} E _ {n} \left ( x +  
 
\frac{k}{m}
 
\frac{k}{m}
Line 105: Line 105:
 
$$  
 
$$  
 
E _ {n} ( mx)  =  -  
 
E _ {n} ( mx)  =  -  
\frac{2 m  ^ {n} }{n+}
+
\frac{2 m  ^ {n} }{n+1}
1
+
\sum _ { k=0} ^ { m-1} ( - 1 )  ^ {k} B _ {n+1}
\sum _ { k= } 0 ^ { m- } 1 ( - 1 )  ^ {k} B _ {n+} 1
 
 
\left ( x +  
 
\left ( x +  
 
\frac{k}{m}
 
\frac{k}{m}
Line 114: Line 113:
  
 
if  $  m $
 
if  $  m $
is even. Here $ B _ {n+} 1 $
+
is even. Here $B_{n+1}$
is a Bernoulli polynomial (cf. [[Bernoulli polynomials|Bernoulli polynomials]]). The periodic functions coinciding with the right-hand side of (*) are extremal in the [[Kolmogorov inequality|Kolmogorov inequality]] and in a number of other extremal problems in function theory. Generalized Euler polynomials have also been considered.
+
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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Euler,  "Opera omnia: series prima: opera mathematica: institutiones calculi differentialis" , Teubner  (1980)  (Translated from Latin)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.E. Nörlund,  "Volesungen über Differenzenrechnung" , Springer  (1924)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  L. Euler,  "Opera omnia: series prima: opera mathematica: institutiones calculi differentialis" , Teubner  (1980)  (Translated from Latin)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  N.E. Nörlund,  "Volesungen über Differenzenrechnung" , Springer  (1924)</TD></TR>
 +
</table>
  
 
====Comments====
 
====Comments====
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  1  
 
  1  
 
\end{array}
 
\end{array}
  \right ) h E _ {n-} 1 ( x) + \dots +
+
  \right ) h E _ {n-1} ( x) + \dots +
 
\left ( \begin{array}{c}
 
\left ( \begin{array}{c}
 
n \\
 
n \\
 
  n- 1  
 
  n- 1  
 
\end{array}
 
\end{array}
  \right ) h  ^ {n-} 1 E _ {1} ( x) + E _ {0} ( x),
+
  \right ) h  ^ {n-1} E _ {1} ( x) + E _ {0} ( x),
 
$$
 
$$
  
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$$
 
$$
  
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 $
+
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} $
 
and then replace  $  \{ E ( x) \}  ^ {i} $
 
with  $  E _ {i} ( x) $.
 
with  $  E _ {i} ( x) $.

Revision as of 08:34, 6 January 2024


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

$$ \tag{* } E _ {n} ( x) = n! over {\pi ^ {n+} 1 } \sum _ { k=0} ^ \infty \frac{\cos [ ( 2 k + 1 ) \pi x + ( n+ 1) \pi / 2 ] }{( 2 k + 1 ) ^ {n+} 1 } , $$

$$ 0 \leq x \leq 1 ,\ n \geq 1 . $$

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) $,

$$ p ( E ( x) + 1) + p( E( x) ) = 2 p( x) . $$

How to Cite This Entry:
Euler polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_polynomials&oldid=46860
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article