Bernoulli number
n | fraction | decimal |
---|---|---|
0 | 1 | +1.000000000 |
1 | ±1/2 | ±0.500000000 |
2 | 1/6 | +0.166666666 |
3 | 0 | +0.000000000 |
4 | −1/30 | −0.033333333 |
5 | 0 | +0.000000000 |
6 | 1/42 | +0.023809523 |
7 | 0 | +0.000000000 |
8 | −1/30 | −0.033333333 |
9 | 0 | +0.000000000 |
10 | 5/66 | +0.075757575 |
11 | 0 | +0.000000000 |
12 | −691/2730 | −0.253113553 |
13 | 0 | +0.000000000 |
14 | 7/6 | +1.166666666 |
15 | 0 | +0.000000000 |
16 | −3617/510 | −7.092156862 |
17 | 0 | +0.000000000 |
18 | 43867/798 | +54.97117794 |
19 | 0 | +0.000000000 |
20 | −174611/330 | −529.1242424 |
In mathematics, the Bernoulli numbers B_{n} are a sequence of rational numbers which occur frequently in number theory. The values of the first 20 Bernoulli numbers are given in the table to the right. For every even n other than 0, B_{n} is negative if n is divisible by 4 and positive otherwise. For every odd n other than 1, B_{n} = 0.
The superscript ± used in this article designates the two sign conventions for Bernoulli numbers. Only the n = 1 term is affected:
- B^{−}
_{n} with B^{−}
_{1} = −1/2 (A027641 / A027642) is the sign convention prescribed by NIST and most modern textbooks.^{[2]} - B^{+}
_{n} with B^{+}
_{1} = +1/2 (A164555 / A027642) is sometimes used in the older literature.^{[3]}
In the formulas below, one can switch from one sign convention to the other with the relation .
The Bernoulli numbers are special values of the Bernoulli polynomials , with and .^{[3]}
Since B_{n} = 0 for all odd n > 1, and many formulas only involve even-index Bernoulli numbers, some authors write "B_{n}" to mean B_{2n}. This article does not follow this notation.
The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of powers of the first positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function.
The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and independently by Japanese mathematician Seki Kōwa. Seki's discovery was posthumously published in 1712^{[4]}^{[5]} in his work Katsuyo Sampo; Bernoulli's, also posthumously, in his Ars Conjectandi of 1713. Ada Lovelace's note G on the Analytical Engine from 1842 describes an algorithm for generating Bernoulli numbers with Babbage's machine.^{[6]} As a result, the Bernoulli numbers have the distinction of being the subject of the first published complex computer program.
History
Early history
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.
Methods to calculate the sum of the first n positive integers, the sum of the squares and of the cubes of the first n positive integers were known, but there were no real 'formulas', only descriptions given entirely in words. Among the great mathematicians of antiquity to consider this problem were Pythagoras (c. 572–497 BCE, Greece), Archimedes (287–212 BCE, Italy), Aryabhata (b. 476, India), Abu Bakr al-Karaji (d. 1019, Persia) and Abu Ali al-Hasan ibn al-Hasan ibn al-Haytham (965–1039, Iraq).
During the late sixteenth and early seventeenth centuries mathematicians made significant progress. 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.
Thomas Harriot seems to have been the first to derive and write formulas for sums of powers using symbolic notation, but even he calculated only up to the sum of the fourth powers. Johann Faulhaber gave formulas for sums of powers up to the 17th power in his 1631 Academia Algebrae, far higher than anyone before him, but he did not give a general formula.
Blaise Pascal in 1654 proved Pascal's identity relating 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 B_{0}, B_{1}, B_{2},… which provide a uniform formula for all sums of powers (Knuth 1993).
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. He wrote:
- "With the help of this table, it took me less than half of a quarter of an hour to find that the tenth powers of the first 1000 numbers being added together will yield the sum 91,409,924,241,424,243,424,241,924,242,500."
Bernoulli's result was published posthumously in Ars Conjectandi in 1713. Seki Kōwa independently discovered the Bernoulli numbers and his result was published a year earlier, also posthumously, in 1712.^{[4]} However, Seki did not present his method as a formula based on a sequence of constants.
Bernoulli's formula for sums of powers is the most useful and generalizable formulation to date. The coefficients in Bernoulli's formula are now called Bernoulli numbers, following a suggestion of Abraham de Moivre.
Bernoulli's formula is sometimes called Faulhaber's formula after Johann Faulhaber who found remarkable ways to calculate sum of powers but never stated Bernoulli's formula. To call Bernoulli's formula Faulhaber's formula does injustice to Bernoulli and simultaneously hides the genius of Faulhaber as Faulhaber's formula is in fact more efficient than Bernoulli's formula. According to Knuth (Knuth 1993) a rigorous proof of Faulhaber's formula was first published by Carl Jacobi in 1834 (Jacobi 1834). Knuth's in-depth study of Faulhaber's formula concludes (the nonstandard notation on the LHS is explained further on):
- "Faulhaber never discovered the Bernoulli numbers; i.e., he never realized that a single sequence of constants B_{0}, B_{1}, B_{2}, … would provide a uniform
- or
- for all sums of powers. He never mentioned, for example, the fact that almost half of the coefficients turned out to be zero after he had converted his formulas for ∑ n^{m} from polynomials in N to polynomials in n." (Knuth 1993, p. 14)
Reconstruction of "Summae Potestatum"
The Bernoulli numbers
This formula suggests setting B_{1} = 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). Most striking in this context is the fact that the falling factorial c^{k−1} has for k = 0 the value 1/c + 1.^{[9]} Thus Bernoulli's formula can be written
if B_{1} = 1/2, recapturing the value Bernoulli gave to the coefficient at that position.
Note that the formula for in the first half contains an error at the last term: it must be instead of .
Definitions
Many characterizations of the Bernoulli numbers have been found in the last 300 years, and each could be used to introduce these numbers. Here only three of the most useful ones are mentioned:
- a recursive equation,
- an explicit formula,
- a generating function.
For the proof of the equivalence of the three approaches see (Ireland & Rosen 1990) or (Conway & Guy 1996).
Recursive definition
The Bernoulli numbers obey the sum formulas ^{[3]}
where and δ denotes the Kronecker delta. Solving for gives the recursive formulas
Explicit definition
In 1893 Louis Saalschütz listed a total of 38 explicit formulas for the Bernoulli numbers (Saalschütz 1893), usually giving some reference in the older literature. One of them is:
Generating function
The exponential generating functions are
The (normal) generating function
is an asymptotic series. It contains the trigamma function ψ_{1}.
Bernoulli numbers and the Riemann zeta function
The Bernoulli numbers can be expressed in terms of the Riemann zeta function:
- B^{+}
_{n} = −nζ(1 − n) for n ≥ 1.
Here the argument of the zeta function is 0 or negative.
By means of the zeta functional equation and the gamma reflection formula the following relation can be obtained:^{[10]}
- for n ≥ 1.
Now the argument of the zeta function is positive.
It then follows from ζ → 1 (n → ∞) and Stirling's formula that
- for n → ∞.
Efficient computation of Bernoulli numbers
In some applications it is useful to be able to compute the Bernoulli numbers B_{0} through B_{p − 3} modulo p, where p is a prime; for example to test whether Vandiver's conjecture holds for p, or even just to determine whether p is an irregular prime. It is not feasible to carry out such a computation using the above recursive formulae, since at least (a constant multiple of) p^{2} arithmetic operations would be required. Fortunately, faster methods have been developed (Buhler et al. 2001) which require only O(p (log p)^{2}) operations (see big O notation).
David Harvey (Harvey 2008) describes an algorithm for computing Bernoulli numbers by computing B_{n} modulo p for many small primes p, and then reconstructing B_{n} via the Chinese remainder theorem. Harvey writes that the asymptotic time complexity of this algorithm is O(n^{2} log(n)^{2 + ε}) and claims that this implementation is significantly faster than implementations based on other methods. Using this implementation Harvey computed B_{n} for n = 10^{8}. Harvey's implementation has been included in SageMath since version 3.1. Prior to that, Bernd Kellner (Kellner 2002) computed B_{n} to full precision for n = 10^{6} in December 2002 and Oleksandr Pavlyk (Pavlyk 2008) for n = 10^{7} with Mathematica in April 2008.
Computer Year n Digits* J. Bernoulli ~1689 10 1 L. Euler 1748 30 8 J. C. Adams 1878 62 36 D. E. Knuth, T. J. Buckholtz 1967 672 1 330 3 G. Fee, S. Plouffe 1996 000 10 677 27 G. Fee, S. Plouffe 1996 000 100 755 376 B. C. Kellner 2002 000000 1 767529 4 O. Pavlyk 2008 000000 10 675260 57 D. Harvey 2008 000000 100 752569 676 - * Digits is to be understood as the exponent of 10 when B_{n} is written as a real number in normalized scientific notation.
Applications of the Bernoulli numbers
Asymptotic analysis
Arguably the most important application of the Bernoulli number in mathematics is their use in the Euler–Maclaurin formula. Assuming that f is a sufficiently often differentiable function the Euler–Maclaurin formula can be written as ^{[11]}
This formulation assumes the convention B^{−}
_{1} = −1/2. Using the convention B^{+}
_{1} = +1/2 the formula becomes
Here f^{(0)} = f (i.e. the zeroth-order derivative of a f is just f). Moreover, let f^{(−1)} denote an antiderivative of f. By the fundamental theorem of calculus,
Thus the last formula can be further simplified to the following succinct form of the Euler–Maclaurin formula
This form is for example the source for the important Euler–Maclaurin expansion of the zeta function
Here s^{k} denotes the rising factorial power.^{[12]}
Bernoulli numbers are also frequently used in other kinds of asymptotic expansions. The following example is the classical Poincaré-type asymptotic expansion of the digamma function ψ.
Sum of powers
Bernoulli numbers feature prominently in the closed form expression of the sum of the mth powers of the first n positive integers. For m, n ≥ 0 define
This expression can always be rewritten as a polynomial in n of degree m + 1. The coefficients of these polynomials are related to the Bernoulli numbers by Bernoulli's formula:
where (^{m + 1}
_{k}) denotes the binomial coefficient.
For example, taking m to be 1 gives the triangular numbers 0, 1, 3, 6, …
Taking m to be 2 gives the square pyramidal numbers 0, 1, 5, 14, …
Some authors use the alternate convention for Bernoulli numbers and state Bernoulli's formula in this way:
Bernoulli's formula is sometimes called Faulhaber's formula after Johann Faulhaber who also found remarkable ways to calculate sums of powers.
Faulhaber's formula was generalized by V. Guo and J. Zeng to a q-analog (Guo & Zeng 2005).
Taylor series
The Bernoulli numbers appear in the Taylor series expansion of many trigonometric functions and hyperbolic functions.
Laurent series
The Bernoulli numbers appear in the following Laurent series:
Digamma function: ^{[13]}
Use in topology
The Kervaire–Milnor's formula for the order of the cyclic group of diffeomorphism classes of exotic (4n − 1)-spheres which bound parallelizable manifolds involves Bernoulli numbers. Let ES_{n} be the number of such exotic spheres for n ≥ 2, then
The Hirzebruch signature theorem for the L genus of a smooth oriented closed manifold of dimension 4n also involves Bernoulli numbers.
Connections with combinatorial numbers
The connection of the Bernoulli number to various kinds of combinatorial numbers is based on the classical theory of finite differences and on the combinatorial interpretation of the Bernoulli numbers as an instance of a fundamental combinatorial principle, the inclusion–exclusion principle.
Connection with Worpitzky numbers
The definition to proceed with was developed by Julius Worpitzky in 1883. Besides elementary arithmetic only the factorial function n! and the power function k^{m} is employed. The signless Worpitzky numbers are defined as
They can also be expressed through the Stirling numbers of the second kind
A Bernoulli number is then introduced as an inclusion–exclusion sum of Worpitzky numbers weighted by the harmonic sequence 1, 1/2, 1/3, …
- B_{0} = 1
- B_{1} = 1 − 1/2
- B_{2} = 1 − 3/2 + 2/3
- B_{3} = 1 − 7/2 + 12/3 − 6/4
- B_{4} = 1 − 15/2 + 50/3 − 60/4 + 24/5
- B_{5} = 1 − 31/2 + 180/3 − 390/4 + 360/5 − 120/6
- B_{6} = 1 − 63/2 + 602/3 − 2100/4 + 3360/5 − 2520/6 + 720/7
This representation has B^{+}
_{1} = +1/2.
Consider the sequence s_{n}, n ≥ 0. From Worpitzky's numbers
Identity of Worpitzky's representation and Akiyama–Tanigawa transform 1 0 1 0 0 1 0 0 0 1 0 0 0 0 1 1 −1 0 2 −2 0 0 3 −3 0 0 0 4 −4 1 −3 2 0 4 −10 6 0 0 9 −21 12 1 −7 12 −6 0 8 −38 54 −24 1 −15 50 −60 24
The first row represents s_{0}, s_{1}, s_{2}, s_{3}, s_{4}.
Hence for the second fractional Euler numbers
- E_{0} = 1
- E_{1} = 1 − 1/2
- E_{2} = 1 − 3/2 + 2/4
- E_{3} = 1 − 7/2 + 12/4 − 6/8
- E_{4} = 1 − 15/2 + 50/4 − 60/8 + 24/16
- E_{5} = 1 − 31/2 + 180/4 − 390/8 + 360/16 − 120/32
- E_{6} = 1 − 63/2 + 602/4 − 2100/8 + 3360/16 − 2520/32 + 720/64
A second formula representing the Bernoulli numbers by the Worpitzky numbers is for n ≥ 1
The simplified second Worpitzky's representation of the second Bernoulli numbers is:
which links the second Bernoulli numbers to the second fractional Euler numbers. The beginning is:
- 1/2, 1/6, 0, −1/30, 0, 1/42, … = (1/2, 1/3, 3/14, 2/15, 5/62, 1/21, …) × (1, 1/2, 0, −1/4, 0, 1/2, …)
The numerators of the first parentheses are
Connection with Stirling numbers of the second kind
If S(k,m) denotes Stirling numbers of the second kind^{[14]} then one has:
where j^{m} denotes the falling factorial.
If one defines the Bernoulli polynomials B_{k}(j) as:^{[15]}
where B_{k} for k = 0, 1, 2,… are the Bernoulli numbers.
Then after the following property of binomial coefficient:
one has,
One also has following for Bernoulli polynomials,^{[15]}
The coefficient of j in (^{j}
_{m + 1}) is (−1)^{m}/m + 1.
Comparing the coefficient of j in the two expressions of Bernoulli polynomials, one has:
(resulting in B_{1} = +1/2) which is an explicit formula for Bernoulli numbers and can be used to prove Von-Staudt Clausen theorem.^{[16]}^{[17]}^{[18]}
Connection with Stirling numbers of the first kind
The two main formulas relating the unsigned Stirling numbers of the first kind [^{n}
_{m}] to the Bernoulli numbers (with B_{1} = +1/2) are
and the inversion of this sum (for n ≥ 0, m ≥ 0)
Here the number A_{n,m} are the rational Akiyama–Tanigawa numbers, the first few of which are displayed in the following table.
Akiyama–Tanigawa number mn0 1 2 3 4 0 1 1/2 1/3 1/4 1/5 1 1/2 1/3 1/4 1/5 … 2 1/6 1/6 3/20 … … 3 0 1/30 … … … 4 −1/30 … … … …
The Akiyama–Tanigawa numbers satisfy a simple recurrence relation which can be exploited to iteratively compute the Bernoulli numbers. This leads to the algorithm shown in the section 'algorithmic description' above. See
An autosequence is a sequence which has its inverse binomial transform equal to the signed sequence. If the main diagonal is zeroes =
Akiyama–Tanigawa transform for the second Euler numbers mn0 1 2 3 4 0 1 1/2 1/4 1/8 1/16 1 1/2 1/2 3/8 1/4 … 2 0 1/4 3/8 … … 3 −1/4 −1/4 … … … 4 0 … … … …
See
- (
A164555 (n + 2)/ A027642 (n + 2) = 1/6, 0, −1/30, 0, 1/42, …) × ( 2^{n + 3} − 2/n + 2 = 3, 14/3, 15/2, 62/5, 21, …) = A198631 (n + 1)/ A006519 (n + 2) = 1/2, 0, −1/4, 0, 1/2, ….
Also valuable for
Connection with Pascal’s triangle
There are formulas connecting Pascal's triangle to Bernoulli numbers ^{[19]}
where is the determinant of a n-by-n square matrix part of Pascal’s triangle whose elements are:
Example:
Connection with Eulerian numbers
There are formulas connecting Eulerian numbers ⟨^{n}
_{m}⟩ to Bernoulli numbers:
Both formulae are valid for n ≥ 0 if B_{1} is set to 1/2. If B_{1} is set to −1/2 they are valid only for n ≥ 1 and n ≥ 2 respectively.
A binary tree representation
The Stirling polynomials σ_{n}(x) are related to the Bernoulli numbers by B_{n} = n!σ_{n}(1). S. C. Woon (Woon 1997) described an algorithm to compute σ_{n}(1) as a binary tree:
Woon's recursive algorithm (for n ≥ 1) starts by assigning to the root node N = [1,2]. Given a node N = [a_{1}, a_{2}, …, a_{k}] of the tree, the left child of the node is L(N) = [−a_{1}, a_{2} + 1, a_{3}, …, a_{k}] and the right child R(N) = [a_{1}, 2, a_{2}, …, a_{k}]. A node N = [a_{1}, a_{2}, …, a_{k}] is written as ±[a_{2}, …, a_{k}] in the initial part of the tree represented above with ± denoting the sign of a_{1}.
Given a node N the factorial of N is defined as
Restricted to the nodes N of a fixed tree-level n the sum of 1/N! is σ_{n}(1), thus
For example:
- B_{1} = 1!(1/2!)
- B_{2} = 2!(−1/3! + 1/2!2!)
- B_{3} = 3!(1/4! − 1/2!3! − 1/3!2! + 1/2!2!2!)
Integral representation and continuation
The integral
has as special values b(2n) = B_{2n} for n > 0.
For example, b(3) = 3/2ζ(3)π^{−3}i and b(5) = −15/2ζ(5)π^{−5}i. Here, ζ is the Riemann zeta function, and i is the imaginary unit. Leonhard Euler (Opera Omnia, Ser. 1, Vol. 10, p. 351) considered these numbers and calculated
The relation to the Euler numbers and π
The Euler numbers are a sequence of integers intimately connected with the Bernoulli numbers. Comparing the asymptotic expansions of the Bernoulli and the Euler numbers shows that the Euler numbers E_{2n} are in magnitude approximately 2/π(4^{2n} − 2^{2n}) times larger than the Bernoulli numbers B_{2n}. In consequence:
This asymptotic equation reveals that π lies in the common root of both the Bernoulli and the Euler numbers. In fact π could be computed from these rational approximations.
Bernoulli numbers can be expressed through the Euler numbers and vice versa. Since, for odd n, B_{n} = E_{n} = 0 (with the exception B_{1}), it suffices to consider the case when n is even.
These conversion formulas express an inverse relation between the Bernoulli and the Euler numbers. But more important, there is a deep arithmetic root common to both kinds of numbers, which can be expressed through a more fundamental sequence of numbers, also closely tied to π. These numbers are defined for n > 1 as
and S_{1} = 1 by convention (Elkies 2003). The magic of these numbers lies in the fact that they turn out to be rational numbers. This was first proved by Leonhard Euler in a landmark paper (Euler 1735) ‘De summis serierum reciprocarum’ (On the sums of series of reciprocals) and has fascinated mathematicians ever since. The first few of these numbers are
These are the coefficients in the expansion of sec x + tan x.
The Bernoulli numbers and Euler numbers are best understood as special views of these numbers, selected from the sequence S_{n} and scaled for use in special applications.
The expression [n even] has the value 1 if n is even and 0 otherwise (Iverson bracket).
These identities show that the quotient of Bernoulli and Euler numbers at the beginning of this section is just the special case of R_{n} = 2S_{n}/S_{n + 1} when n is even. The R_{n} are rational approximations to π and two successive terms always enclose the true value of π. Beginning with n = 1 the sequence starts (
These rational numbers also appear in the last paragraph of Euler's paper cited above.
Consider the Akiyama–Tanigawa transform for the sequence
0 1 1/2 0 −1/4 −1/4 −1/8 0 1 1/2 1 3/4 0 −5/8 −3/4 2 −1/2 1/2 9/4 5/2 5/8 3 −1 −7/2 −3/4 15/2 4 5/2 −11/2 −99/4 5 8 77/2 6 −61/2
From the second, the numerators of the first column are the denominators of Euler's formula. The first column is −1/2 ×
An algorithmic view: the Seidel triangle
The sequence S_{n} has another unexpected yet important property: The denominators of S_{n} divide the factorial (n − 1)!. In other words: the numbers T_{n} = S_{n}(n − 1)!, sometimes called Euler zigzag numbers, are integers.
Thus the above representations of the Bernoulli and Euler numbers can be rewritten in terms of this sequence as
These identities make it easy to compute the Bernoulli and Euler numbers: the Euler numbers E_{n} are given immediately by T_{2n + 1} and the Bernoulli numbers B_{2n} are obtained from T_{2n} by some easy shifting, avoiding rational arithmetic.
What remains is to find a convenient way to compute the numbers T_{n}. However, already in 1877 Philipp Ludwig von Seidel (Seidel 1877) published an ingenious algorithm which makes it extremely simple to calculate T_{n}.
- Start by putting 1 in row 0 and let k denote the number of the row currently being filled
- If k is odd, then put the number on the left end of the row k − 1 in the first position of the row k, and fill the row from the left to the right, with every entry being the sum of the number to the left and the number to the upper
- At the end of the row duplicate the last number.
- If k is even, proceed similar in the other direction.
Seidel's algorithm is in fact much more general (see the exposition of Dominique Dumont (Dumont 1981)) and was rediscovered several times thereafter.
Similar to Seidel's approach D. E. Knuth and T. J. Buckholtz (Knuth & Buckholtz 1967) gave a recurrence equation for the numbers T_{2n} and recommended this method for computing B_{2n} and E_{2n} ‘on electronic computers using only simple operations on integers’.
V. I. Arnold rediscovered Seidel's algorithm in (Arnold 1991) and later Millar, Sloane and Young popularized Seidel's algorithm under the name boustrophedon transform.
Triangular form:
1 1 1 2 2 1 2 4 5 5 16 16 14 10 5 16 32 46 56 61 61 272 272 256 224 178 122 61
Only
Distribution with a supplementary 1 and one 0 in the following rows:
1 0 1 −1 −1 0 0 −1 −2 −2 5 5 4 2 0 0 5 10 14 16 16 −61 −61 −56 −46 −32 −16 0
This is
The Akiyama–Tanigawa algorithm applied to
1 1 1/2 0 −1/4 −1/4 −1/8 0 1 3/2 1 0 −3/4 −1 −1 3/2 4 15/4 0 −5 −15/2 1 5 5 −51/2 0 61 −61
1. The first column is
1 1 0 −2 0 16 0 0 −1 −2 2 16 −16 −1 −1 4 14 −32 0 5 10 −46 5 5 −56 0 −61 −61
The first row of this array is
2. The second column is 1 1 −1 −5 5 61 −61 −1385 1385…. Its binomial transform yields:
1 2 2 −4 −16 32 272 1 0 −6 −12 48 240 −1 −6 −6 60 192 −5 0 66 32 5 66 66 61 0 −61
The first row of this array is 1 2 2 −4 −16 32 272 544 −7936 15872 353792 −707584…. The absolute values of the second bisection are the double of the absolute values of the first bisection.
Consider the Akiyama-Tanigawa algorithm applied to
1 2 2 3/2 1 3/4 3/4 −1 0 3/2 2 5/4 0 −1 −3 −3/2 3 25/4 2 −3 −27/2 −13 5 21 −3/2 −16 45 −61
The first column whose the absolute values are
0 −1 −1 2 5 −16 −61 −1 0 3 3 −21 −45 1 3 0 −24 −24 2 −3 −24 0 −5 −21 24 −16 45 −61
The first two upper diagonals are −1 3 −24 402… = (−1)^{n + 1} ×
−
2 1 −1 −2 5 16 −61 −1 −2 −1 7 11 −77 −1 1 8 4 −88 2 7 −4 −92 5 −11 −88 −16 −77 −61
The main diagonal, here 2 −2 8 −92…, is the double of the first upper one, here
A combinatorial view: alternating permutations
Around 1880, three years after the publication of Seidel's algorithm, Désiré André proved a now classic result of combinatorial analysis (André 1879) & (André 1881). Looking at the first terms of the Taylor expansion of the trigonometric functions tan x and sec x André made a startling discovery.
The coefficients are the Euler numbers of odd and even index, respectively. In consequence the ordinary expansion of tan x + sec x has as coefficients the rational numbers S_{n}.
André then succeeded by means of a recurrence argument to show that the alternating permutations of odd size are enumerated by the Euler numbers of odd index (also called tangent numbers) and the alternating permutations of even size by the Euler numbers of even index (also called secant numbers).
Related sequences
The arithmetic mean of the first and the second Bernoulli numbers are the associate Bernoulli numbers:
B_{0} = 1, B_{1} = 0, B_{2} = 1/6, B_{3} = 0, B_{4} = −1/30,
The Akiyama–Tanigawa algorithm applied to
1 5/6 3/4 7/10 2/3 1/6 1/6 3/20 2/15 5/42 0 1/30 1/20 2/35 5/84 −1/30 −1/30 −3/140 −1/105 0 0 −1/42 −1/28 −4/105 −1/28
Hence another link between the intrinsic Bernoulli numbers and the Balmer series via
The terms of the first row are f(n) = 1/2 + 1/n + 2. 2, f(n) is an autosequence of the second kind. 3/2, f(n) leads by its inverse binomial transform to 3/2 −1/2 1/3 −1/4 1/5 ... = 1/2 + log 2.
Consider g(n) = 1/2 - 1 / (n+2) = 0, 1/6, 1/4, 3/10, 1/3. The Akiyama-Tanagiwa transforms gives:
0 1/6 1/4 3/10 1/3 5/14 ... −1/6 −1/6 −3/20 −2/15 −5/42 −3/28 ... 0 −1/30 −1/20 −2/35 −5/84 −5/84 ... 1/30 1/30 3/140 1/105 0 −1/140 ...
0, g(n), is an autosequence of the second kind.
Euler
1 1 7/8 3/4 21/32 0 1/4 3/8 3/8 5/16 −1/4 −1/4 0 1/4 25/64 0 −1/2 −3/4 −9/16 −5/32 1/2 1/2 −9/16 −13/8 −125/64
The first line is Eu(n). Eu(n) preceded by a zero is an autosequence of the first kind. It is linked to the Oresme numbers. The numerators of the second line are
0 1 1 7/8 3/4 21/32 19/32 1 0 −1/8 −1/8 −3/32 −1/16 −5/128 −1 −1/8 0 1/32 1/32 3/128 1/64
Generalization to the odd-index Bernoulli numbers
The Bernoulli numbers B_{n} for n = 0, 1, 2, 3, ... are
- 1, 1/2, 1/6, 3/56, 1/30, 25/992, 1/42, 427/16256, 1/30, 12465/261632, 5/66, 555731/4102256, 691/2730, 35135945/67100672, 7/6, 2990414715/1073709056, ... (
A193472/ A193473)^{[20]}
This is ez(n − 1)n!/4^{n} − 2^{n} where ez(n) is the nth coefficient of sec t + tan t (
A companion to the Bernoulli numbers
See
A191754 / A192366 = 0, 1/2, 1/2, 1/3, 1/6, 1/15, 1/30, 1/35, 1/70, –1/105, –1/210, 41/1155, 41/2310, –589/5005, −589/10010 …
Apply T(n + 1, k) = 2T(n, k + 1) − T(n,k) to T(0,k) =
0 1/2 1/2 1/3 1/6 1/15 1 1/2 1/6 0 −1/30 0 0 −1/6 −1/6 −1/15 1/30 1/21 −1/3 −1/6 1/30 2/15 13/210 −2/21 0 7/30 7/30 −1/105 −53/210 −13/105 7/15 7/30 −53/210 −52/105 1/210 92/105
The rows are alternatively autosequences of the first and of the second kind. The second row is
The first column is 0, 1, 0, −1/3, 0, 7/15, 0, −31/21, 0, 127/105, 0, −511/33,… from Mersenne primes, see
Consider the triangle
0 1 0 1 1 0 1 2 1 0
This is Pascal's triangle
0 1 1 1 1 0 1 2 3 4 0 1 3 6 10 0 1 4 10 20 0 1 5 15 35
Multiplying the SBD array in
0 1/2 1/2 0 1/2 −1/6 1/2 −2/6 0 1/2 −3/6 1/15 1/2 −4/6 3/15 0 1/2 −5/6 6/15 −4/105
This triangle is unreduced.
Arithmetical properties of the Bernoulli numbers
The Bernoulli numbers can be expressed in terms of the Riemann zeta function as B_{n} = −nζ(1 − n) for integers n ≥ 0 provided for n = 0 and n = 1 the expression −nζ(1 − n) is understood as the limiting value and the convention B_{1} = 1/2 is used. This intimately relates them to the values of the zeta function at negative integers. As such, they could be expected to have and do have deep arithmetical properties. For example, the Agoh–Giuga conjecture postulates that p is a prime number if and only if pB_{p − 1} is congruent to −1 modulo p. Divisibility properties of the Bernoulli numbers are related to the ideal class groups of cyclotomic fields by a theorem of Kummer and its strengthening in the Herbrand-Ribet theorem, and to class numbers of real quadratic fields by Ankeny–Artin–Chowla.
The Kummer theorems
The Bernoulli numbers are related to Fermat's Last Theorem (FLT) by Kummer's theorem (Kummer 1850), which says:
- If the odd prime p does not divide any of the numerators of the Bernoulli numbers B_{2}, B_{4}, …, B_{p − 3} then x^{p} + y^{p} + z^{p} = 0 has no solutions in nonzero integers.
Prime numbers with this property are called regular primes. Another classical result of Kummer (Kummer 1851) are the following congruences.
- Let p be an odd prime and b an even number such that p − 1 does not divide b. Then for any non-negative integer k
A generalization of these congruences goes by the name of p-adic continuity.
p-adic continuity
If b, m and n are positive integers such that m and n are not divisible by p − 1 and m ≡ n mod p^{b − 1} (p − 1), then
Since B_{n} = −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 ζ_{p}(s) for all p-adic integers ℤ_{p}, the p-adic zeta function.
Ramanujan's congruences
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:
Von Staudt–Clausen theorem
The von Staudt–Clausen theorem was given by Karl Georg Christian von Staudt (von Staudt 1840) and Thomas Clausen (Clausen 1840) independently in 1840. The theorem states that for every n > 0,
is an integer. The sum extends over all primes p for which p − 1 divides 2n.
A consequence of this is that the denominator of B_{2n} is given by the product of all primes p for which p − 1 divides 2n. In particular, these denominators are square-free and divisible by 6.
Why do the odd Bernoulli numbers vanish?
The sum
can be evaluated for negative values of the index n. Doing so will show that it is an odd function for even values of k, which implies that the sum has only terms of odd index. This and the formula for the Bernoulli sum imply that B_{2k + 1 − m} is 0 for m even and 2k + 1 − m > 1; and that the term for B_{1} is cancelled by the subtraction. The von Staudt–Clausen theorem combined with Worpitzky's representation also gives a combinatorial answer to this question (valid for n > 1).
From the von Staudt–Clausen theorem it is known that for odd n > 1 the number 2B_{n} is an integer. This seems trivial if one knows beforehand that the integer in question is zero. However, by applying Worpitzky's representation one gets
as a sum of integers, which is not trivial. Here a combinatorial fact comes to surface which explains the vanishing of the Bernoulli numbers at odd index. Let S_{n,m} be the number of surjective maps from {1, 2, …, n} to {1, 2, …, m}, then S_{n,m} = m!{^{n}
_{m}}. The last equation can only hold if
This equation can be proved by induction. The first two examples of this equation are
- n = 4: 2 + 8 = 7 + 3,
- n = 6: 2 + 120 + 144 = 31 + 195 + 40.
Thus the Bernoulli numbers vanish at odd index because some non-obvious combinatorial identities are embodied in the Bernoulli numbers.
A restatement of the Riemann hypothesis
The connection between the Bernoulli numbers and the Riemann zeta function is strong enough to provide an alternate formulation of the Riemann hypothesis (RH) which uses only the Bernoulli number. In fact Marcel Riesz (Riesz 1916) proved that the RH is equivalent to the following assertion:
- For every ε > 1/4 there exists a constant C_{ε} > 0 (depending on ε) such that |R(x)| < C_{ε}x^{ε} as x → ∞.
Here R(x) is the Riesz function
n^{k} denotes the rising factorial power in the notation of D. E. Knuth. The numbers β_{n} = B_{n}/n occur frequently in the study of the zeta function and are significant because β_{n} is a p-integer for primes p where p − 1 does not divide n. The β_{n} are called divided Bernoulli numbers.
Generalized Bernoulli numbers
The generalized Bernoulli numbers are certain algebraic numbers, defined similarly to the Bernoulli numbers, that are related to special values of Dirichlet L-functions in the same way that Bernoulli numbers are related to special values of the Riemann zeta function.
Let χ be a Dirichlet character modulo f. The generalized Bernoulli numbers attached to χ are defined by
Apart from the exceptional B_{1,1} = 1/2, we have, for any Dirichlet character χ, that B_{k,χ} = 0 if χ(−1) ≠ (−1)^{k}.
Generalizing the relation between Bernoulli numbers and values of the Riemann zeta function at non-positive integers, one has the for all integers k ≥ 1:
where L(s,χ) is the Dirichlet L-function of χ.^{[21]}
Appendix
Assorted identities
- Umbral calculus gives a compact form of Bernoulli's formula by using an abstract symbol B:
where the symbol B^{k} that appears during binomial expansion of the parenthesized term is to be replaced by the Bernoulli number B_{k} (and B_{1} = +1/2). More suggestively and mnemonically, this may be written as a definite integral:
Many other Bernoulli identities can be written compactly with this symbol, e.g.
- Let n be non-negative and even
- The nth cumulant of the uniform probability distribution on the interval [−1, 0] is B_{n}/n.
- Let n? = 1/n! and n ≥ 1. Then B_{n} is the following (n + 1) × (n + 1) determinant:^{[22]}
- For even-numbered Bernoulli numbers, B_{2p} is given by the (p + 1) × (p + 1) determinant:^{[22]}
- Let n ≥ 1. Then (Leonhard Euler)
- Let n ≥ 1. Then (von Ettingshausen 1827)
- Let n ≥ 0. Then (Leopold Kronecker 1883)
- Let n ≥ 1 and m ≥ 1. Then (Carlitz 1968)
- Let n ≥ 4 and
the harmonic number. Then (H. Miki 1978)
- Let n ≥ 4. Yuri Matiyasevich found (1997)
- Faber–Pandharipande–Zagier–Gessel identity: for n ≥ 1,
- The next formula is true for n ≥ 0 if B_{1} = B_{1}(1) = 1/2, but only for n ≥ 1 if B_{1} = B_{1}(0) = −1/2.
- Let n ≥ 0. Then
and
- A reciprocity relation of M. B. Gelfand (Agoh & Dilcher 2008):
See also
- Bernoulli polynomial
- Bernoulli polynomials of the second kind
- Euler number
- Genocchi number
- Kummer's congruences
- Poly-Bernoulli number
- Hurwitz zeta function
- Euler summation
- Stirling polynomial
- Sums of powers
Notes
- ↑ "Bernoulli Number". Wolfram MathWorld. Retrieved 14 August 2016.
- ↑ Arfken, George (1970). Mathematical methods for physicists (2nd edition). Academic Press, Inc. p. 278. ISBN 978-0120598519.
- 1 2 3 "Bernoulli Number". Wolfram MathWorld. Retrieved 2 July 2017.
- 1 2 Selin, H. (1997), p. 891
- ↑ Smith, D. E. (1914), p. 108
- ↑ Note G in the Menabrea reference
- ↑ Mathematics Genealogy Project
- ↑ Earliest Uses of Symbols of Calculus
- ↑ Graham, R.; Knuth, D. E.; Patashnik, O. (1989), Concrete Mathematics (2nd ed.), Addison-Wesley, Section 2.51, ISBN 0-201-55802-5
- ↑ Arfken, George (1970). Mathematical methods for physicists (2nd edition). Academic Press, Inc. p. 279. ISBN 978-0120598519.
- ↑ Concrete Mathematics, (9.67).
- ↑ Concrete Mathematics, (2.44) and (2.52)
- ↑ Arfken, George (1970). Mathematical methods for physicists (2nd edition). Academic Press, Inc. p. 463. ISBN 978-0120598519.
- ↑ L. Comtet, Advanced combinatorics. The art of finite and infinite expansions, Revised and Enlarged Edition, D. Reidel Publ. Co., Dordrecht-Boston, 1974.
- 1 2 H. Rademacher, Analytic Number Theory, Springer-Verlag, New York, 1973.
- ↑ H. W. Gould (1972). "Explicit formulas for Bernoulli numbers". Amer. Math. Monthly. 79: 44–51. doi:10.2307/2978125.
- ↑ T. M. Apostol. Introduction to Analytic Number Theory. Springer-Verlag. p. 197.
- ↑ G. Boole (1880). A treatise of the calculus of finite differences (3rd ed.). London.
- ↑ this formula was discovered (or perhaps rediscovered) by Giorgio Pietrocola. His demonstration is available in Italian language: Pietrocola, Giorgio (October 31, 2008). "Esplorando un antico sentiero: teoremi sulla somma di potenze di interi successivi (Corollario 2b)". Maecla. Retrieved April 8, 2017.
- ↑ Odd-index Bernoulli numbers
- ↑ Neukirch 1999, §VII.2
- 1 2 Jerome Malenfant (2011). "Finite, closed-form expressions for the partition function and for Euler, Bernoulli, and Stirling numbers". arXiv:1103.1585 [math.NT].
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External links
- Hazewinkel, Michiel, ed. (2001) [1994], "Bernoulli numbers", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- The first 498 Bernoulli Numbers from Project Gutenberg
- A multimodular algorithm for computing Bernoulli numbers
- The Bernoulli Number Page
- Bernoulli number programs at LiteratePrograms
- Weisstein, Eric W. "Bernoulli Number". MathWorld.
- P. Luschny. "The Computation of Irregular Primes".
- P. Luschny. "The Computation And Asymptotics Of Bernoulli Numbers".
- Gottfried Helms. "Bernoullinumbers in context of Pascal-(Binomial)matrix" (PDF).
- Gottfried Helms. "summing of like powers in context with Pascal-/Bernoulli-matrix" (PDF).
- Gottfried Helms. "Some special properties, sums of Bernoulli-and related numbers" (PDF).