View source for Bernoulli polynomials

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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|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 [[#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

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

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|Appell polynomials]], i.e. they satisfy the condition

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

and are closely connected with the [[Euler polynomials|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|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|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|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====
<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>