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

$${x\over e^x-1} + {x\over 2} -1 \equiv \sum_{n=1}^\infty {(-1)^{n-1}B_n^* x^{2n}\over (2n)!}$$

$$= {B_1^* x^2\over 2!} - {B_2^* x^4\over 4!} + {B_3^* x^6\over 6!} + \ldots$$ (1)

for $\vert x\vert < 2\pi$, or

$$1 - {x\over 2} \cot\left({x\over 2}\right) \equiv \sum_{n=1}^\infty {B_n^* x^{2n}\over (2n)!}$$

$$= {B_1^* x^2\over 2!} + {B_2^* x^4\over 4!} + {B_3^* x^6\over 6!} + \ldots$$ (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

$$B_n^* = 4n\int_0^\infty {t^{2n-1}\,dt\over e^{2\pi t}-1},$$ (4)

and analytically from

$$B_n^* = {2(2n)!\over (2\pi)^{2n}} \sum_{p=1}^\infty p^{-2n} = {2(2n)!\over (2\pi)^{2n}} \zeta(2n)$$ (5)

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

The first few Bernoulli numbers $B_n^*$ are

$$B_1^* = {\textstyle{1\over 6}}$$

$$B_2^* = {\textstyle{1\over 30}}$$

$$B_3^* = {\textstyle{1\over 42}}$$

$$B_4^* = {\textstyle{1\over 30}}$$

$$B_5^* = {\textstyle{5\over 66}}$$

$$B_6^* = {\textstyle{691\over 2{,}730}}$$

$$B_7^* = {\textstyle{7\over 6}}$$

$$B_8^* = {\textstyle{3{,}617\over 510}}$$

$$B_9^* = {\textstyle{43{,}867\over 798}}$$

$$B_{10}^* = {\textstyle{174{,}611\over 330}}$$

$$B_{11}^* = {\textstyle{854{,}513\over 138}}.$$

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

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

The $B_n$ can be defined by the identity

$${x\over e^x-1} \equiv\sum_{n=0}^\infty {B_n x^n\over n!}.$$ (7)

These relationships can be derived using the generating function

$$F(x,t)=\sum_{n=0}^\infty {B_n(x)t^n\over n!},$$ (8)

which converges uniformly for $\vert t\vert<2\pi$ and all $x$ (Castellanos 1988). Taking the partial derivative gives

$${\partial F(x,t)\over\partial x}=\sum_{n=0}^\infty {B_{n-1}(... ...\over(n-1)!}=t\sum_{n=0}^\infty {B_n(x)t^n\over n!} = tF(x,t).$$ (9)

The solution to this differential equation is

$$F(x,t)=T(t)e^{xt},$$ (10)

so integrating gives

$$\int_0^1 F(x,t)\,dx = T(t)\int_0^1 e^{xt}\,dx = T(t){e^t-1\over t}$$

$$= \sum_{n=0}^\infty {t^n\over n!}\int_0^1 B_n(x)\,dx$$

$$= 1+\sum_{n=1}^\infty {t^n\over n!} \int_0^1 B_n(x)\,dx=1$$ (11)

or

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

(Castellanos 1988). Setting $x=0$ and adding $t/2$ to both sides then gives

$${\textstyle{1\over 2}}t\coth({\textstyle{1\over 2}}t)=\sum_{n=0}^\infty {B_{2n}t^{2n}\over(2n)!}.$$ (13)

Letting $t=2ix$ then gives

$$x \cot x = \sum_{n=0}^\infty (-1)^nB_{2n} {{2x}^{2n}\over (2n)!}$$ (14)

for $x \in [-\pi,\pi]$. The Bernoulli numbers may also be calculated from the integral

$$B_n = {n!\over 2\pi i} \int {z\over e^z-1} {dz\over z^{n+1}},$$ (15)

or from

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

The Bernoulli numbers satisfy the identity

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

where ${n\choose k}$ is a Binomial Coefficient. An asymptotic Formula is

$$\lim_{n\to\infty} \vert B_{2n}\vert \sim 4\sqrt{\pi n}\left({n\over \pi e}\right)^{2n}.$$ (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,

$$B_{2n} = {(-1)^{n-1}2(2n)!\over (2\pi)^{2n}} \sum_{p=1}^\infty p^{-2n} = {(-1)^{n-1}2(2n)!\over (2\pi)^{2n}}\zeta(2n)$$ (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

$$B_n = (-1)^{n+1}n\zeta(1-n).$$ (20)

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

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

$$B_0 = 1$$

$$B_1 = -{\textstyle{1\over 2}}$$

$$B_2 = {\textstyle{1\over 6}}$$

$$B_4 = -{\textstyle{1\over 30}}$$

$$B_6 = {\textstyle{1\over 42}}$$

$$B_8 = -{\textstyle{1\over 30}}$$

$$B_{10} = {\textstyle{5\over 66}}$$

$$B_{12} = -{\textstyle{691\over 2{,}730}}$$

$$B_{14} = {\textstyle{7\over 6}}$$

$$B_{16} = -{\textstyle{3{,}617\over 510}}$$

$$B_{18} = {\textstyle{43{,}867\over 798}}$$

$$B_{20} = -{\textstyle{174{,}611\over 330}}$$

$$B_{22} = {\textstyle{854{,}513\over 138}}$$

(Sloane's A000367 and A002445). In addition,

$$B_{2n+1} = 0$$ (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

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

$$\sum_{k=1}^n k^p = {(B+n+1)^{[p+1]}-B^{p+1}\over p+1},$$ (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,

$$\sum_{k=1}^n k^p = \sum_{k=1}^n {B_k p!\over k!(p-k+1)!} n^{p-k+1}.$$ (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

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.

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.'' http://www.lacim.uqam.ca/pi/records.html.

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.'' http://www.research.att.com/~njas/sequences/eisonline.html 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.