Bernoulli polynomials

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