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Bernoulli Polynomial


There are two definitions of Bernoulli polynomials in use. The $n$th Bernoulli polynomial is denoted here by $B_n(x)$, and the archaic Bernoulli polynomial by $B_n^*(x)$. These definitions correspond to the Bernoulli Numbers evaluated at 0,

$\displaystyle B_n$ $\textstyle \equiv$ $\displaystyle B_n(0)$ (1)
$\displaystyle B_n^*$ $\textstyle \equiv$ $\displaystyle B_n^*(0).$ (2)

They also satisfy
B_n(1) = (-1)^nB_n(0)
\end{displaymath} (3)

\end{displaymath} (4)

(Lehmer 1988). The first few Bernoulli Polynomials are
$\displaystyle B_0(x)$ $\textstyle =$ $\displaystyle 1$  
$\displaystyle B_1(x)$ $\textstyle =$ $\displaystyle x -{\textstyle{1\over 2}}$  
$\displaystyle B_2(x)$ $\textstyle =$ $\displaystyle x^2-x + {\textstyle{1\over 6}}$  
$\displaystyle B_3(x)$ $\textstyle =$ $\displaystyle x^3 - {\textstyle{3\over 2}} x^2 + {\textstyle{1\over 2}}x$  
$\displaystyle B_4(x)$ $\textstyle =$ $\displaystyle x^4-2x^3+x^2 - {\textstyle{1\over 30}}$  
$\displaystyle B_5(x)$ $\textstyle =$ $\displaystyle x^5- {\textstyle{5\over 2}} x^4 + {\textstyle{5\over 3}}x^3 - {\textstyle{1\over 6}} x$  
$\displaystyle B_6(x)$ $\textstyle =$ $\displaystyle x^6-3x^5+ {\textstyle{5\over 2}}x^4 - {\textstyle{1\over 2}}x^2 + {\textstyle{1\over 42}}.$  

Bernoulli (1713) defined the polynomials in terms of sums of the Powers of consecutive integers,
\sum_{k=0}^{m-1} k^{n-1}={1\over n}[B_n(m)-B_n(0)].
\end{displaymath} (5)

Euler (1738) gave the Bernoulli polynomials $B_n(x)$ in terms of the generating function
{te^{tx}\over e^t-1} \equiv \sum_{n=0}^\infty B_n(x) {t^n\over n!}.
\end{displaymath} (6)

They satisfy the Recurrence Relation
{dB_n\over dx} = nB_{n-1}(x)
\end{displaymath} (7)

(Appell 1882), and obey the identity
\end{displaymath} (8)

where $B^k$ is interpreted here as $B_k(x)$. Hurwitz gave the Fourier Series
B_n(x)=-{n!\over(2\pi i)^n}\sum_{k=-\infty}^\infty k^{-n}e^{2\pi ikx},
\end{displaymath} (9)

for $0<x<1$, and Raabe (1851) found
{1\over m}\sum_{k=0}^{m-1} B_n\left({x+{k\over m}}\right)=m^{-n}B_n(mx).
\end{displaymath} (10)

A sum identity involving the Bernoulli Polynomials is

\sum_{k=0}^m {m\choose k} B_k(\alpha) B_{m-k}(\beta) = -(m-1)B_m(\alpha+\beta)+m(\alpha+\beta-1)B_{m-1}(\alpha+\beta)
\end{displaymath} (11)

for an Integer $m$ and arbitrary Real Numbers $\alpha$ and $\beta$.

See also Bernoulli Number, Euler-Maclaurin Integration Formulas, Euler Polynomial


Abramowitz, M. and Stegun, C. A. (Eds.). ``Bernoulli and Euler Polynomials and the Euler-Maclaurin Formula.'' §23.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 804-806, 1972.

Appell, P. E. ``Sur une classe de polynomes.'' Annales d'École Normal Superieur, Ser. 2 9, 119-144, 1882.

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 330, 1985.

Bernoulli, J. Ars conjectandi. Basel, Switzerland, p. 97, 1713. Published posthumously.

Euler, L. ``Methodus generalis summandi progressiones.'' Comment. Acad. Sci. Petropol. 6, 68-97, 1738.

Lehmer, D. H. ``A New Approach to Bernoulli Polynomials.'' Amer. Math. Monthly. 95, 905-911, 1988.

Lucas, E. Ch. 14 in Théorie des Nombres. Paris, 1891.

Raabe, J. L. ``Zurückführung einiger Summen und bestimmten Integrale auf die Jakob Bernoullische Function.'' J. reine angew. Math. 42, 348-376, 1851.

Spanier, J. and Oldham, K. B. ``The Bernoulli Polynomial $B_n(x)$.'' Ch. 19 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 167-173, 1987.

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© 1996-9 Eric W. Weisstein