Bernoulli Polynomial

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There are two definitions of Bernoulli polynomials in use. The th Bernoulli polynomial is denoted here by , and the archaic Bernoulli polynomial by . 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

$\begin{displaymath} B_n(1) = (-1)^nB_n(0) \end{displaymath}$

(3)

and

$\begin{displaymath} B_n(1-x)=(-1)^nB_n(x) \end{displaymath}$

(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,

$\begin{displaymath} \sum_{k=0}^{m-1} k^{n-1}={1\over n}[B_n(m)-B_n(0)]. \end{displaymath}$

(5)

Euler (1738) gave the Bernoulli polynomials

in terms of the generating function

$\begin{displaymath} {te^{tx}\over e^t-1} \equiv \sum_{n=0}^\infty B_n(x) {t^n\over n!}. \end{displaymath}$

(6)

They satisfy the Recurrence Relation

$\begin{displaymath} {dB_n\over dx} = nB_{n-1}(x) \end{displaymath}$

(7)

(Appell 1882), and obey the identity

$\begin{displaymath} B_n(x)=(B+x)^n, \end{displaymath}$

(8)

where

is interpreted here as

. Hurwitz gave the Fourier Series

$\begin{displaymath} B_n(x)=-{n!\over(2\pi i)^n}\sum_{k=-\infty}^\infty k^{-n}e^{2\pi ikx}, \end{displaymath}$

(9)

for

, and Raabe (1851) found

$\begin{displaymath} {1\over m}\sum_{k=0}^{m-1} B_n\left({x+{k\over m}}\right)=m^{-n}B_n(mx). \end{displaymath}$

(10)

A sum identity involving the Bernoulli Polynomials is

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

(11)

for an Integer

and arbitrary Real Numbers $\alpha$ and $\beta$ .

References

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 .'' Ch. 19 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 167-173, 1987.