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Bernoulli Number

There are two definitions for the Bernoulli numbers. The older one, no longer in widespread use, defines the Bernoulli numbers $B_n^*$ by the equations

$\displaystyle {x\over e^x-1} + {x\over 2} -1$ $\textstyle \equiv$ $\displaystyle \sum_{n=1}^\infty {(-1)^{n-1}B_n^* x^{2n}\over (2n)!}$  
  $\textstyle =$ $\displaystyle {B_1^* x^2\over 2!} - {B_2^* x^4\over 4!} + {B_3^* x^6\over 6!} + \ldots$ (1)

for $\vert x\vert < 2\pi$, or
$\displaystyle 1 - {x\over 2} \cot\left({x\over 2}\right)$ $\textstyle \equiv$ $\displaystyle \sum_{n=1}^\infty {B_n^* x^{2n}\over (2n)!}$  
  $\textstyle =$ $\displaystyle {B_1^* x^2\over 2!} + {B_2^* x^4\over 4!} + {B_3^* x^6\over 6!} + \ldots$ (3)

for $\vert x\vert < \pi$ (Whittaker and Watson 1990, p. 125). Gradshteyn and Ryzhik (1979) denote these numbers $B_n^*$, while Bernoulli numbers defined by the newer (National Bureau of Standards) definition are denoted $B_n$. The $B_n^*$ Bernoulli numbers may be calculated from the integral
B_n^* = 4n\int_0^\infty {t^{2n-1}\,dt\over e^{2\pi t}-1},
\end{displaymath} (4)

and analytically from
B_n^* = {2(2n)!\over (2\pi)^{2n}} \sum_{p=1}^\infty p^{-2n} = {2(2n)!\over (2\pi)^{2n}} \zeta(2n)
\end{displaymath} (5)

for $n=1$, 2, ..., where $\zeta(z)$ is the Riemann Zeta Function.

The first few Bernoulli numbers $B_n^*$ are

$\displaystyle B_1^*$ $\textstyle =$ $\displaystyle {\textstyle{1\over 6}}$  
$\displaystyle B_2^*$ $\textstyle =$ $\displaystyle {\textstyle{1\over 30}}$  
$\displaystyle B_3^*$ $\textstyle =$ $\displaystyle {\textstyle{1\over 42}}$  
$\displaystyle B_4^*$ $\textstyle =$ $\displaystyle {\textstyle{1\over 30}}$  
$\displaystyle B_5^*$ $\textstyle =$ $\displaystyle {\textstyle{5\over 66}}$  
$\displaystyle B_6^*$ $\textstyle =$ $\displaystyle {\textstyle{691\over 2{,}730}}$  
$\displaystyle B_7^*$ $\textstyle =$ $\displaystyle {\textstyle{7\over 6}}$  
$\displaystyle B_8^*$ $\textstyle =$ $\displaystyle {\textstyle{3{,}617\over 510}}$  
$\displaystyle B_9^*$ $\textstyle =$ $\displaystyle {\textstyle{43{,}867\over 798}}$  
$\displaystyle B_{10}^*$ $\textstyle =$ $\displaystyle {\textstyle{174{,}611\over 330}}$  
$\displaystyle B_{11}^*$ $\textstyle =$ $\displaystyle {\textstyle{854{,}513\over 138}}.$  

Bernoulli numbers defined by the modern definition are denoted $B_n$ and also called ``Even-index'' Bernoulli numbers. These are the Bernoulli numbers returned by the Mathematica ${}^{\scriptstyle\circledRsymbol}$ (Wolfram Research, Champaign, IL) function BernoulliB[n]. These Bernoulli numbers are a superset of the archaic ones $B_n^*$ since

1 & for $n=0$\cr
-{\textstyle{1\over 2}}&...
...)^{(n/2)-1}B_{n/2}^* & for $n$\ even\cr
0 & for $n$\ odd.\cr}
\end{displaymath} (6)

The $B_n$ can be defined by the identity
{x\over e^x-1} \equiv\sum_{n=0}^\infty {B_n x^n\over n!}.
\end{displaymath} (7)

These relationships can be derived using the generating function
F(x,t)=\sum_{n=0}^\infty {B_n(x)t^n\over n!},
\end{displaymath} (8)

which converges uniformly for $\vert t\vert<2\pi$ and all $x$ (Castellanos 1988). Taking the partial derivative gives
{\partial F(x,t)\over\partial x}=\sum_{n=0}^\infty {B_{n-1}(...
...\over(n-1)!}=t\sum_{n=0}^\infty {B_n(x)t^n\over n!} = tF(x,t).
\end{displaymath} (9)

The solution to this differential equation is
\end{displaymath} (10)

so integrating gives
$\displaystyle \int_0^1 F(x,t)\,dx$ $\textstyle =$ $\displaystyle T(t)\int_0^1 e^{xt}\,dx = T(t){e^t-1\over t}$  
  $\textstyle =$ $\displaystyle \sum_{n=0}^\infty {t^n\over n!}\int_0^1 B_n(x)\,dx$  
  $\textstyle =$ $\displaystyle 1+\sum_{n=1}^\infty {t^n\over n!} \int_0^1 B_n(x)\,dx=1$ (11)

{te^{xt}\over e^t-1}=\sum_{n=0}^\infty {B_n(x)t^n\over n!}
\end{displaymath} (12)

(Castellanos 1988). Setting $x=0$ and adding $t/2$ to both sides then gives
{\textstyle{1\over 2}}t\coth({\textstyle{1\over 2}}t)=\sum_{n=0}^\infty {B_{2n}t^{2n}\over(2n)!}.
\end{displaymath} (13)

Letting $t=2ix$ then gives
x \cot x = \sum_{n=0}^\infty (-1)^nB_{2n} {{2x}^{2n}\over (2n)!}
\end{displaymath} (14)

for $x \in [-\pi,\pi]$. The Bernoulli numbers may also be calculated from the integral
B_n = {n!\over 2\pi i} \int {z\over e^z-1} {dz\over z^{n+1}},
\end{displaymath} (15)

or from
B_n = \left[{{d^n\over dx^n} {x\over e^x-1}}\right]_{x=0}.
\end{displaymath} (16)

The Bernoulli numbers satisfy the identity

{k+1\choose 1}B_k+{k+1\choose 2}B_{k-1}+\ldots+{k+1\choose k}B_1+B_0=0,
\end{displaymath} (17)

where ${n\choose k}$ is a Binomial Coefficient. An asymptotic Formula is
\lim_{n\to\infty} \vert B_{2n}\vert \sim 4\sqrt{\pi n}\left({n\over \pi e}\right)^{2n}.
\end{displaymath} (18)

Bernoulli numbers appear in expressions of the form $\sum_{k=1}^n k^p$, where $p = 1$, 2, .... Bernoulli numbers also appear in the series expansions of functions involving $\tan x$, $\cot x$, $\csc x$, $\ln\!\vert\sin x\vert$, $\ln\!\vert\cos x\vert$, $\ln\!\vert\tan x\vert$, $\tanh x$, $\coth x$, and $\mathop{\rm csch}\nolimits x$. An analytic solution exists for Even orders,
B_{2n} = {(-1)^{n-1}2(2n)!\over (2\pi)^{2n}} \sum_{p=1}^\infty p^{-2n} = {(-1)^{n-1}2(2n)!\over (2\pi)^{2n}}\zeta(2n)
\end{displaymath} (19)

for $n=1$, 2, ..., where $\zeta(2n)$ is the Riemann Zeta Function. Another intimate connection with the Riemann Zeta Function is provided by the identity
B_n = (-1)^{n+1}n\zeta(1-n).
\end{displaymath} (20)

The Denominator of $B_{2k}$ is given by the von Staudt-Clausen Theorem

\mathop{\rm denom}(B_{2k})=\prod_{\scriptstyle p{\rm\ prime}\atop\scriptstyle (p-1)\vert 2k}^{2k+1} p,
\end{displaymath} (21)

which also implies that the Denominator of $B_{2k}$ is Squarefree (Hardy and Wright 1979). Another curious property is that the fraction part of $B_n$ in Decimal has a Decimal Period which divides $n$, and there is a single digit before that period (Conway 1996).
$\displaystyle B_0$ $\textstyle =$ $\displaystyle 1$  
$\displaystyle B_1$ $\textstyle =$ $\displaystyle -{\textstyle{1\over 2}}$  
$\displaystyle B_2$ $\textstyle =$ $\displaystyle {\textstyle{1\over 6}}$  
$\displaystyle B_4$ $\textstyle =$ $\displaystyle -{\textstyle{1\over 30}}$  
$\displaystyle B_6$ $\textstyle =$ $\displaystyle {\textstyle{1\over 42}}$  
$\displaystyle B_8$ $\textstyle =$ $\displaystyle -{\textstyle{1\over 30}}$  
$\displaystyle B_{10}$ $\textstyle =$ $\displaystyle {\textstyle{5\over 66}}$  
$\displaystyle B_{12}$ $\textstyle =$ $\displaystyle -{\textstyle{691\over 2{,}730}}$  
$\displaystyle B_{14}$ $\textstyle =$ $\displaystyle {\textstyle{7\over 6}}$  
$\displaystyle B_{16}$ $\textstyle =$ $\displaystyle -{\textstyle{3{,}617\over 510}}$  
$\displaystyle B_{18}$ $\textstyle =$ $\displaystyle {\textstyle{43{,}867\over 798}}$  
$\displaystyle B_{20}$ $\textstyle =$ $\displaystyle -{\textstyle{174{,}611\over 330}}$  
$\displaystyle B_{22}$ $\textstyle =$ $\displaystyle {\textstyle{854{,}513\over 138}}$  

(Sloane's A000367 and A002445). In addition,
B_{2n+1} = 0
\end{displaymath} (22)

for $n=1$, 2, ....

Bernoulli first used the Bernoulli numbers while computing $\sum_{k=1}^n k^p$. He used the property of the Figurate Number Triangle that

\sum_{i=0}^n a_{ij} = {(n+1)a_{nj}\over j+1},
\end{displaymath} (23)

along with a form for $a_{nj}$ which he derived inductively to compute the sums up to $n=10$ (Boyer 1968, p. 85). For $p
\in \Bbb{Z} > 0$, the sum is given by
\sum_{k=1}^n k^p = {(B+n+1)^{[p+1]}-B^{p+1}\over p+1},
\end{displaymath} (24)

where the Notation $B^{[k]}$ means the quantity in question is raised to the appropriate Power $k$, and all terms of the form $B^m$ are replaced with the corresponding Bernoulli numbers $B_m$. Written explicitly in terms of a sum of Powers,
\sum_{k=1}^n k^p = \sum_{k=1}^n {B_k p!\over k!(p-k+1)!} n^{p-k+1}.
\end{displaymath} (25)

It is also true that the Coefficients of the terms in such an expansion sum to 1 (which Bernoulli stated without proof). Ramanujan gave a number of curious infinite sum identities involving Bernoulli numbers (Berndt 1994).

G. J. Fee and S. Plouffe have computed $B_{200,000}$, which has $\sim 800,000$ Digits (Plouffe). Plouffe and collaborators have also calculated $B_n$ for $n$ up to 72,000.

See also Argoh's Conjecture, Bernoulli Function, Bernoulli Polynomial, Debye Functions, Euler-Maclaurin Integration Formulas, Euler Number, Figurate Number Triangle, Genocchi Number, Pascal's Triangle, Riemann Zeta Function, von Staudt-Clausen Theorem


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.

Arfken, G. ``Bernoulli Numbers, Euler-Maclaurin Formula.'' §5.9 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 327-338, 1985.

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 71, 1987.

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 81-85, 1994.

Boyer, C. B. A History of Mathematics. New York: Wiley, 1968.

Castellanos, D. ``The Ubiquitous Pi. Part I.'' Math. Mag. 61, 67-98, 1988.

Conway, J. H. and Guy, R. K. In The Book of Numbers. New York: Springer-Verlag, pp. 107-110, 1996.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, 1980.

Hardy, G. H. and Wright, W. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Oxford University Press, pp. 91-93, 1979.

Ireland, K. and Rosen, M. ``Bernoulli Numbers.'' Ch. 15 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 228-248, 1990.

Knuth, D. E. and Buckholtz, T. J. ``Computation of Tangent, Euler, and Bernoulli Numbers.'' Math. Comput. 21, 663-688, 1967.

Plouffe, S. ``Plouffe's Inverter: Table of Current Records for the Computation of Constants.''

Ramanujan, S. ``Some Properties of Bernoulli's Numbers.'' J. Indian Math. Soc. 3, 219-234, 1911.

Sloane, N. J. A. Sequences A000367/M4039 and A002445/M4189 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Spanier, J. and Oldham, K. B. ``The Bernoulli Numbers, $B_n$.'' Ch. 4 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 35-38, 1987.

Wagstaff, S. S. Jr. ``Ramanujan's Paper on Bernoulli Numbers.'' J. Indian Math. Soc. 45, 49-65, 1981.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.

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