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Difference between revisions of "Bernoulli polynomials"

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The polynomials
 
The polynomials
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b0156501.png" /></td> </tr></table>
+
$$
 +
B _ {n} (x)  = \
 +
\sum _ { s=0 } ^ { n }
 +
\left ( \begin{array}{c}
 +
n \\
 +
s
 +
\end{array}
 +
\right )
 +
B _ {s} x  ^ {n-s} \ \
 +
(n = 0, 1 ,\dots ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b0156502.png" /> are the [[Bernoulli numbers|Bernoulli numbers]]. Thus, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b0156503.png" />
+
where $  B _ {s} $
 +
are the [[Bernoulli numbers|Bernoulli numbers]]. Thus, for $  n = 0, 1, 2, 3, $
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b0156504.png" /></td> </tr></table>
+
$$
 +
B _ {0} (x)  = 1,\ \
 +
B _ {1} (x)  = \
 +
x -
 +
\frac{1}{2}
 +
,\ \
 +
B _ {2} (x)  = \
 +
x  ^ {2} - x + {
 +
\frac{1}{6}
 +
} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b0156505.png" /></td> </tr></table>
+
$$
 +
B _ {3} (x)  = x  ^ {3} -  
 +
\frac{3}{2}
 +
x  ^ {2} +
 +
\frac{1}{2}
 +
x.
 +
$$
  
 
Bernoulli polynomials may be computed by the recurrence formula
 
Bernoulli polynomials may be computed by the recurrence formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b0156506.png" /></td> </tr></table>
+
$$
 +
\sum _ { s=0 } ^ { n-1 }
 +
\left ( \begin{array}{c}
 +
n \\
 +
s
 +
\end{array}
 +
\right )
 +
B _ {s} (x)  = \
 +
nx  ^ {n-1} ,\ \
 +
n = 2, 3 ,\dots .
 +
$$
  
For a natural argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b0156507.png" /> Bernoulli polynomials were first studied by Jacob Bernoulli in 1713, in connection with the computation of the sum
+
For a natural argument $  x = m $
 +
Bernoulli polynomials were first studied by Jacob Bernoulli in 1713, in connection with the computation of the sum
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b0156508.png" /></td> </tr></table>
+
$$
 +
\sum _ { k=0 } ^ { m }  k  ^ {n} .
 +
$$
  
L. Euler [[#References|[1]]] was the first to study Bernoulli polynomials for arbitrary values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b0156509.png" />. The term  "Bernoulli polynomials"  was introduced by J.L. Raabe in 1851. The fundamental property of such polynomials is that they satisfy the finite-difference equation
+
L. Euler [[#References|[1]]] was the first to study Bernoulli polynomials for arbitrary values of $  x $.  
 +
The term  "Bernoulli polynomials"  was introduced by J.L. Raabe in 1851. The fundamental property of such polynomials is that they satisfy the finite-difference equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565010.png" /></td> </tr></table>
+
$$
 +
B _ {n} (x+1) - B _ {n} (x)  = \
 +
n x  ^ {n-1} ,
 +
$$
  
 
and therefore play the same role in finite-difference calculus as do power functions in differential calculus.
 
and therefore play the same role in finite-difference calculus as do power functions in differential calculus.
Line 25: Line 81:
 
Bernoulli polynomials belong to the class of [[Appell polynomials|Appell polynomials]], i.e. they satisfy the condition
 
Bernoulli polynomials belong to the class of [[Appell polynomials|Appell polynomials]], i.e. they satisfy the condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565011.png" /></td> </tr></table>
+
$$
 +
B _ {n} ^ { \prime } (x)  = nB _ {n-1} (x)
 +
$$
  
 
and are closely connected with the [[Euler polynomials|Euler polynomials]]
 
and are closely connected with the [[Euler polynomials|Euler polynomials]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565012.png" /></td> </tr></table>
+
$$
 +
E _ {n-1} (x)  =
 +
\frac{2}{n}
 +
 
 +
\left [
 +
B _ {n} (x) -2  ^ {n} B _ {n} \left (
 +
\frac{x}{2}
 +
\right )  \right ] .
 +
$$
  
 
The generating function of the Bernoulli polynomials is
 
The generating function of the Bernoulli polynomials is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565013.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{te  ^ {tx} }{e  ^ {t} -1 }
 +
  = \
 +
\sum _ { n=0 } ^  \infty 
 +
 
 +
\frac{B _ {n} (x) }{n!}
 +
t  ^ {n} .
 +
$$
 +
 
 +
Bernoulli polynomials are expandable into [[Fourier series|Fourier series]]: For  $  n = 1 $
 +
 
 +
$$
 +
B _ {1} (x)  = \
 +
x -  
 +
\frac{1}{2}
 +
  = -
 +
\sum _ { s=1 } ^  \infty 
  
Bernoulli polynomials are expandable into [[Fourier series|Fourier series]]: For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565014.png" />
+
\frac{\sin  2 \pi sx }{s \pi }
 +
,\ \
 +
0 < x < 1 ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565015.png" /></td> </tr></table>
+
and for  $  n = 2, 3 \dots $
  
and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565016.png" />
+
$$
 +
B _ {n} (x)  = - 2n!
 +
\sum _ { s=1 } ^  \infty 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565017.png" /></td> </tr></table>
+
\frac{\cos \left ( 2 \pi sx -  
 +
\frac{n\pi}{2}
 +
\right ) }{( 2 \pi s)  ^ {n} }
 +
,\ \
 +
0 \leq  x \leq  1 .
 +
$$
  
 
Bernoulli polynomials satisfy the relationships
 
Bernoulli polynomials satisfy the relationships
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565018.png" /></td> </tr></table>
+
$$
 +
B _ {n} (mx)  = \
 +
m  ^ {n-1}
 +
\sum _ { s=0 } ^ { m-1 }
 +
B _ {n} \left ( x +
 +
\frac{s}{m}
 +
\right )
 +
$$
  
 
(the multiplication theorem);
 
(the multiplication theorem);
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565019.png" /></td> </tr></table>
+
$$
 +
B _ {n} (1-x)  = \
 +
(-1)  ^ {n} B _ {n} (x)
 +
$$
  
 
(the complement theorem);
 
(the complement theorem);
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565020.png" /></td> </tr></table>
+
$$
 +
B _ {n} (x+y)  = \
 +
\sum _ { s=0 } ^ { n }
 +
\left ( \begin{array}{c}
 +
n \\
 +
s
 +
\end{array}
 +
\right )
 +
B _ {s} (y) x  ^ {n-s}
 +
$$
  
 
(the theorem on addition of arguments).
 
(the theorem on addition of arguments).
Line 59: Line 171:
 
Bernoulli polynomials are employed to express the residual term of the [[Euler–MacLaurin formula|Euler–MacLaurin formula]], and for the expansion of functions into series. Many important properties of Bernoulli numbers are a consequence of the properties of Bernoulli polynomials. Bernoulli polynomials are employed in the integral representation of differentiable periodic functions
 
Bernoulli polynomials are employed to express the residual term of the [[Euler–MacLaurin formula|Euler–MacLaurin formula]], and for the expansion of functions into series. Many important properties of Bernoulli numbers are a consequence of the properties of Bernoulli polynomials. Bernoulli polynomials are employed in the integral representation of differentiable periodic functions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565021.png" /></td> </tr></table>
+
$$
 +
\psi (x)  =
 +
\frac{1}{2 \pi }
 +
 
 +
\int\limits _ {- \pi } ^  \pi 
 +
f(t)  dt +
 +
\frac{1} \pi
 +
 
 +
\int\limits _ {- \pi } ^  \pi 
 +
\phi _ {k} (t) f  ^ {(k-1)}
 +
(x + \pi - t )  dt ,
 +
$$
 +
 
 +
$$
 +
\phi _ {k} (t)  =
 +
\frac{2  ^ {k-1} \pi  ^ {k} }{k!}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565022.png" /></td> </tr></table>
+
B _ {k} \left (
 +
\frac{\pi +t }{2 \pi }
 +
\right ) ,
 +
$$
  
 
and play an important part in the theory of approximation of such functions by trigonometric polynomials and other aggregates. Cf. [[Favard problem|Favard problem]].
 
and play an important part in the theory of approximation of such functions by trigonometric polynomials and other aggregates. Cf. [[Favard problem|Favard problem]].
  
Various generalizations of the Bernoulli polynomials have been proposed. N.E. Nörlund introduced generalized Bernoulli polynomials of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565023.png" /> and degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565024.png" />:
+
Various generalizations of the Bernoulli polynomials have been proposed. N.E. Nörlund introduced generalized Bernoulli polynomials of order $  \nu $
 +
and degree $  n $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565025.png" /></td> </tr></table>
+
$$
 +
B _ {n} ^ { ( \nu ) } (x \mid  \omega )  = \
 +
B _ {n} ^ { ( \nu ) } (x \mid  \omega _ {1} \dots \omega _ {v} )
 +
$$
  
 
(certain special cases of such polynomials had been previously considered by V.G. Immenetskii, N.Ya. Sonin and D.M. Sintsov). Let
 
(certain special cases of such polynomials had been previously considered by V.G. Immenetskii, N.Ya. Sonin and D.M. Sintsov). Let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565026.png" /></td> </tr></table>
+
$$
 +
\Delta _  \omega  f (x)  = \
 +
 
 +
\frac{f(x + \omega )-f(x) } \omega
 +
 
 +
$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565027.png" /></td> </tr></table>
+
$$
 +
B _ {n} ^ { (0) } (x \mid  \omega )  = x  ^ {n} ,\ \
 +
n = 0, 1 ,\dots ;
 +
$$
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565028.png" /> is successively determined as the polynomial solution of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565029.png" /> of the finite-difference equation
+
then $  B _ {n} ^ { ( \nu ) } (x \mid  \omega ) $
 +
is successively determined as the polynomial solution of degree $  n $
 +
of the finite-difference equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565030.png" /></td> </tr></table>
+
$$
 +
\Delta _ {\omega _  \nu  } B _ {n} ^ { (0) } (x \mid  \omega _ {1} \dots
 +
\omega _  \nu  )  = \
 +
nB _ {n-1} ^ { (v-1) }
 +
(x \mid  \omega _ {1} \dots \omega _ {\nu -1 }  ) ,
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565031.png" /> with initial conditions
+
$  \nu = 1, 2 \dots $
 +
with initial conditions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565032.png" /></td> </tr></table>
+
$$
 +
B _ {n} ^ { ( \nu ) } (0 \mid  \omega _ {1} \dots \omega _  \nu  )  = \
 +
B _ {n} ^ { ( \nu ) }
 +
[ \omega _ {1} \dots \omega _  \nu  ] ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565033.png" /> (a generalized Bernoulli number) is found from the recurrence relations
+
where $  B _ {n} ^ { ( \nu ) } [ \omega _ {1} \dots \omega _  \nu  ] $(
 +
a generalized Bernoulli number) is found from the recurrence relations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565034.png" /></td> </tr></table>
+
$$
 +
\sum _ { s=1 } ^ { n }
 +
\left ( \begin{array}{c}
 +
n \\
 +
s
 +
\end{array}
 +
\right )
 +
\omega _  \nu  ^ {s}
 +
B _ {n-s} ^ { ( \nu ) }
 +
= \omega _  \nu  ^ {n}
 +
B _ {n-1} ^ { ( \nu -1) }
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015650/b01565035.png" /></td> </tr></table>
+
$$
 +
(B _ {n} ^ { (1) } [ \omega _ {1} ]  = \omega _ {1}  ^ {n} B _ {n} ,\  B _ {0} ^ { ( \nu ) }  = 1 ,\  B _ {n} ^
 +
{ (0) }  = 0 ,\  B _ {0} ^ { ( \nu ) }  = 1 ) .
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Euler,  "Institutiones calculi differentialis"  G. Kowalewski (ed.) , ''Opera Omnia Ser. 1; opera mat.'' , '''10''' , Teubner  (1980)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.E. Nörlund,  "Volesungen über Differenzenrechnung" , Springer  (1924)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.) , ''Higher transcendental functions'' , '''1. The gamma function. The hypergeometric functions. Legendre functions''' , McGraw-Hill  (1953)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.V. Likhin,  ''Istor Mat. Issled.''  (1959)  pp. 59–134</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Euler,  "Institutiones calculi differentialis"  G. Kowalewski (ed.) , ''Opera Omnia Ser. 1; opera mat.'' , '''10''' , Teubner  (1980)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.E. Nörlund,  "Volesungen über Differenzenrechnung" , Springer  (1924)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.) , ''Higher transcendental functions'' , '''1. The gamma function. The hypergeometric functions. Legendre functions''' , McGraw-Hill  (1953)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.V. Likhin,  ''Istor Mat. Issled.''  (1959)  pp. 59–134</TD></TR></table>

Latest revision as of 10:58, 29 May 2020


The polynomials

$$ B _ {n} (x) = \ \sum _ { s=0 } ^ { n } \left ( \begin{array}{c} n \\ s \end{array} \right ) B _ {s} x ^ {n-s} \ \ (n = 0, 1 ,\dots ), $$

where $ B _ {s} $ are the Bernoulli numbers. Thus, for $ n = 0, 1, 2, 3, $

$$ B _ {0} (x) = 1,\ \ B _ {1} (x) = \ x - \frac{1}{2} ,\ \ B _ {2} (x) = \ x ^ {2} - x + { \frac{1}{6} } , $$

$$ B _ {3} (x) = x ^ {3} - \frac{3}{2} x ^ {2} + \frac{1}{2} x. $$

Bernoulli polynomials may be computed by the recurrence formula

$$ \sum _ { s=0 } ^ { n-1 } \left ( \begin{array}{c} n \\ s \end{array} \right ) B _ {s} (x) = \ nx ^ {n-1} ,\ \ n = 2, 3 ,\dots . $$

For a natural argument $ x = m $ Bernoulli polynomials were first studied by Jacob Bernoulli in 1713, in connection with the computation of the sum

$$ \sum _ { k=0 } ^ { m } k ^ {n} . $$

L. Euler [1] was the first to study Bernoulli polynomials for arbitrary values of $ x $. The term "Bernoulli polynomials" was introduced by J.L. Raabe in 1851. The fundamental property of such polynomials is that they satisfy the finite-difference equation

$$ B _ {n} (x+1) - B _ {n} (x) = \ n x ^ {n-1} , $$

and therefore play the same role in finite-difference calculus as do power functions in differential calculus.

Bernoulli polynomials belong to the class of Appell polynomials, i.e. they satisfy the condition

$$ B _ {n} ^ { \prime } (x) = nB _ {n-1} (x) $$

and are closely connected with the Euler polynomials

$$ E _ {n-1} (x) = \frac{2}{n} \left [ B _ {n} (x) -2 ^ {n} B _ {n} \left ( \frac{x}{2} \right ) \right ] . $$

The generating function of the Bernoulli polynomials is

$$ \frac{te ^ {tx} }{e ^ {t} -1 } = \ \sum _ { n=0 } ^ \infty \frac{B _ {n} (x) }{n!} t ^ {n} . $$

Bernoulli polynomials are expandable into Fourier series: For $ n = 1 $

$$ B _ {1} (x) = \ x - \frac{1}{2} = - \sum _ { s=1 } ^ \infty \frac{\sin 2 \pi sx }{s \pi } ,\ \ 0 < x < 1 , $$

and for $ n = 2, 3 \dots $

$$ B _ {n} (x) = - 2n! \sum _ { s=1 } ^ \infty \frac{\cos \left ( 2 \pi sx - \frac{n\pi}{2} \right ) }{( 2 \pi s) ^ {n} } ,\ \ 0 \leq x \leq 1 . $$

Bernoulli polynomials satisfy the relationships

$$ B _ {n} (mx) = \ m ^ {n-1} \sum _ { s=0 } ^ { m-1 } B _ {n} \left ( x + \frac{s}{m} \right ) $$

(the multiplication theorem);

$$ B _ {n} (1-x) = \ (-1) ^ {n} B _ {n} (x) $$

(the complement theorem);

$$ B _ {n} (x+y) = \ \sum _ { s=0 } ^ { n } \left ( \begin{array}{c} n \\ s \end{array} \right ) B _ {s} (y) x ^ {n-s} $$

(the theorem on addition of arguments).

Bernoulli polynomials are employed to express the residual term of the Euler–MacLaurin formula, and for the expansion of functions into series. Many important properties of Bernoulli numbers are a consequence of the properties of Bernoulli polynomials. Bernoulli polynomials are employed in the integral representation of differentiable periodic functions

$$ \psi (x) = \frac{1}{2 \pi } \int\limits _ {- \pi } ^ \pi f(t) dt + \frac{1} \pi \int\limits _ {- \pi } ^ \pi \phi _ {k} (t) f ^ {(k-1)} (x + \pi - t ) dt , $$

$$ \phi _ {k} (t) = \frac{2 ^ {k-1} \pi ^ {k} }{k!} B _ {k} \left ( \frac{\pi +t }{2 \pi } \right ) , $$

and play an important part in the theory of approximation of such functions by trigonometric polynomials and other aggregates. Cf. Favard problem.

Various generalizations of the Bernoulli polynomials have been proposed. N.E. Nörlund introduced generalized Bernoulli polynomials of order $ \nu $ and degree $ n $:

$$ B _ {n} ^ { ( \nu ) } (x \mid \omega ) = \ B _ {n} ^ { ( \nu ) } (x \mid \omega _ {1} \dots \omega _ {v} ) $$

(certain special cases of such polynomials had been previously considered by V.G. Immenetskii, N.Ya. Sonin and D.M. Sintsov). Let

$$ \Delta _ \omega f (x) = \ \frac{f(x + \omega )-f(x) } \omega $$

and

$$ B _ {n} ^ { (0) } (x \mid \omega ) = x ^ {n} ,\ \ n = 0, 1 ,\dots ; $$

then $ B _ {n} ^ { ( \nu ) } (x \mid \omega ) $ is successively determined as the polynomial solution of degree $ n $ of the finite-difference equation

$$ \Delta _ {\omega _ \nu } B _ {n} ^ { (0) } (x \mid \omega _ {1} \dots \omega _ \nu ) = \ nB _ {n-1} ^ { (v-1) } (x \mid \omega _ {1} \dots \omega _ {\nu -1 } ) , $$

$ \nu = 1, 2 \dots $ with initial conditions

$$ B _ {n} ^ { ( \nu ) } (0 \mid \omega _ {1} \dots \omega _ \nu ) = \ B _ {n} ^ { ( \nu ) } [ \omega _ {1} \dots \omega _ \nu ] , $$

where $ B _ {n} ^ { ( \nu ) } [ \omega _ {1} \dots \omega _ \nu ] $( a generalized Bernoulli number) is found from the recurrence relations

$$ \sum _ { s=1 } ^ { n } \left ( \begin{array}{c} n \\ s \end{array} \right ) \omega _ \nu ^ {s} B _ {n-s} ^ { ( \nu ) } = \omega _ \nu ^ {n} B _ {n-1} ^ { ( \nu -1) } $$

$$ (B _ {n} ^ { (1) } [ \omega _ {1} ] = \omega _ {1} ^ {n} B _ {n} ,\ B _ {0} ^ { ( \nu ) } = 1 ,\ B _ {n} ^ { (0) } = 0 ,\ B _ {0} ^ { ( \nu ) } = 1 ) . $$

References

[1] L. Euler, "Institutiones calculi differentialis" G. Kowalewski (ed.) , Opera Omnia Ser. 1; opera mat. , 10 , Teubner (1980)
[2] N.E. Nörlund, "Volesungen über Differenzenrechnung" , Springer (1924)
[3] H. Bateman (ed.) A. Erdélyi (ed.) , Higher transcendental functions , 1. The gamma function. The hypergeometric functions. Legendre functions , McGraw-Hill (1953)
[4] V.V. Likhin, Istor Mat. Issled. (1959) pp. 59–134
How to Cite This Entry:
Bernoulli polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bernoulli_polynomials&oldid=19249
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article