Bernoulli number

The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function.

As a result, the Bernoulli numbers have the distinction of being the subject of the first published complex computer program.

The Bernoulli numbers are rooted in the early history of the computation of sums of integer powers, which have been of interest to mathematicians since antiquity.

In the West Thomas Harriot (1560–1621) of England, Johann Faulhaber (1580–1635) of Germany, Pierre de Fermat (1601–1665) and fellow French mathematician Blaise Pascal (1623–1662) all played important roles.

Blaise Pascal in 1654 proved Pascal's identity relating (n+1)k+1 to the sums of the pth powers of the first n positive integers for p = 0, 1, 2, ..., k. The Swiss mathematician Jakob Bernoulli (1654–1705) was the first to realize the existence of a single sequence of constants B0, B1, B2,... which provide a uniform formula for all sums of powers.

[9] The joy Bernoulli experienced when he hit upon the pattern needed to compute quickly and easily the coefficients of his formula for the sum of the cth powers for any positive integer c can be seen from his comment.

Seki Takakazu independently discovered the Bernoulli numbers and his result was published a year earlier, also posthumously, in 1712.

The integral symbol on the left hand side goes back to Gottfried Wilhelm Leibniz in 1675 who used it as a long letter S for "summa" (sum).

[b] The letter n on the left hand side is not an index of summation but gives the upper limit of the range of summation which is to be understood as 1, 2, ..., n. Putting things together, for positive c, today a mathematician is likely to write Bernoulli's formula as: This formula suggests setting B1 = ⁠1/2⁠ when switching from the so-called 'archaic' enumeration which uses only the even indices 2, 4, 6... to the modern form (more on different conventions in the next paragraph).

It is not feasible to carry out such a computation using the above recursive formulae, since at least (a constant multiple of) p2 arithmetic operations would be required.

Fortunately, faster methods have been developed[18] which require only O(p (log p)2) operations (see big O notation).

Prior to that, Bernd Kellner[20] computed Bn to full precision for n = 106 in December 2002 and Oleksandr Pavlyk[21] for n = 107 with Mathematica in April 2008.

The following example is the classical Poincaré-type asymptotic expansion of the digamma function ψ. Bernoulli numbers feature prominently in the closed form expression of the sum of the mth powers of the first n positive integers.

The Kervaire–Milnor formula for the order of the cyclic group of diffeomorphism classes of exotic (4n − 1)-spheres which bound parallelizable manifolds involves Bernoulli numbers.

In consequence: This asymptotic equation reveals that π lies in the common root of both the Bernoulli and the Euler numbers.

This was first proved by Leonhard Euler in a landmark paper De summis serierum reciprocarum (On the sums of series of reciprocals) and has fascinated mathematicians ever since.

These identities show that the quotient of Bernoulli and Euler numbers at the beginning of this section is just the special case of Rn = ⁠2Sn/Sn + 1⁠ when n is even.

However, already in 1877 Philipp Ludwig von Seidel published an ingenious algorithm, which makes it simple to calculate Tn.

Around 1880, three years after the publication of Seidel's algorithm, Désiré André proved a now classic result of combinatorial analysis.

[38][39] Looking at the first terms of the Taylor expansion of the trigonometric functions tan x and sec x André made a startling discovery.

In consequence the ordinary expansion of tan x + sec x has as coefficients the rational numbers Sn.

If b, m and n are positive integers such that m and n are not divisible by p − 1 and m ≡ n (mod pb − 1 (p − 1)), then Since Bn = −nζ(1 − n), this can also be written where u = 1 − m and v = 1 − n, so that u and v are nonpositive and not congruent to 1 modulo p − 1.

This tells us that the Riemann zeta function, with 1 − p−s taken out of the Euler product formula, is continuous in the p-adic numbers on odd negative integers congruent modulo p − 1 to a particular a ≢ 1 mod (p − 1), and so can be extended to a continuous function ζp(s) for all p-adic integers

The following relations, due to Ramanujan, provide a method for calculating Bernoulli numbers that is more efficient than the one given by their original recursive definition: The von Staudt–Clausen theorem was given by Karl Georg Christian von Staudt[42] and Thomas Clausen[43] independently in 1840.

The von Staudt–Clausen theorem combined with Worpitzky's representation also gives a combinatorial answer to this question (valid for n > 1).

Here a combinatorial fact comes to surface which explains the vanishing of the Bernoulli numbers at odd index.

But last year I took a close look at Peter Luschny's Bernoulli manifesto, where he gives more than a dozen good reasons why the value of $B_1$ should really be plus one-half.

He explains that some mathematicians of the early 20th century had unilaterally changed the conventions, because some of their formulas came out a bit nicer when the negative value was used.

Even worse, all the major software systems for symbolic mathematics have that 20th-century aberration deeply embedded.

Yet Luschny convinced me that we have all been wrong, and that it's high time to change back to the correct definition before the situation gets even worse.