Bernoulli number
n | |
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0 | |
1 | |
2 | |
3 | 0 |
4 | |
5 | 0 |
6 | |
7 | 0 |
8 | |
9 | 0 |
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11 | 0 |
12 | |
13 | 0 |
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15 | 0 |
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17 | 0 |
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19 | 0 |
20 |
In mathematics, the Bernoulli numbers B_{n} are a sequence of rational numbers with deep connections to number theory. The values of the first few Bernoulli numbers are
- B_{0} = 1, B_{1} = ±Template:Frac, B_{2} = Template:Frac, B_{3} = 0, B_{4} = −Template:Frac, B_{5} = 0, B_{6} = Template:Frac, B_{7} = 0, B_{8} = −Template:Frac.
If the convention B_{1} = −Template:Frac is used, this sequence is also known as the first Bernoulli numbers ( A027641 / A027642 in OEIS); with the convention B_{1} = +Template:Frac is known as the second Bernoulli numbers ( A164555 / A027642). Except for this one difference, the first and second Bernoulli numbers agree. Since B_{n} = 0 for all odd n > 1, and many formulas only involve even-index Bernoulli numbers, some authors write B_{n} instead of B_{2n}.
The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in formulas 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 Jakob Bernoulli, after whom they are named, and independently by Japanese mathematician Seki Kōwa. Seki's discovery was posthumously published in 1712^{[1]}^{[2]} 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.^{[3]} As a result, the Bernoulli numbers have the distinction of being the subject of one of the first computer programs.
Sum of powers
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Bernoulli numbers feature prominently in the closed form expression of the sum of the m-th 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 the convention B_{1} = +1/2 is used. ( denotes the binomial coefficient, m+1 choose k.)
For example, taking m to be 1 gives the triangular numbers 0, 1, 3, 6, ... A000217.
Taking m to be 2 gives the square pyramidal numbers 0, 1, 5, 14, ... A000330.
Some authors use the convention B_{1} = −1/2 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 sum of powers.
Faulhaber's formula was generalized by V. Guo and J. Zeng to a q-analog Template:Harv.
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 four of the most useful ones are mentioned:
- a recursive equation,
- an explicit formula,
- a generating function,
- an algorithmic description.
For the proof of the equivalence of the four approaches the reader is referred to mathematical expositions like Template:Harv or Template:Harv.
Unfortunately in the literature the definition is given in two variants: Despite the fact that Bernoulli defined B_{1} = 1/2 (now known as "second Bernoulli numbers"), some authors set B_{1} = −1/2 ("first Bernoulli numbers"). In order to prevent potential confusions both variants will be described here, side by side. Because these two definitions can be transformed simply by into the other, some formulae have this alternatingly (-1)^{n}-term and others not depending on the context, but it is not possible to decide in favor of one of these definitions to be the correct or appropriate or natural one (for the abstract Bernoulli numbers).
Recursive definition
The recursive equation is best introduced in a slightly more general form
This defines polynomials B_{m} in the variable n known as the Bernoulli polynomials. The recursion can also be viewed as defining rational numbers B_{m}(n) for all integers n ≥ 0, m ≥ 0. The expression 0^{0} has to be interpreted as 1. The first and second Bernoulli numbers now follow by setting n = 0 (resulting in B_{1}=−Template:Frac, "first Bernoulli numbers") respectively n = 1 (resulting in B_{1}=+Template:Frac, "second Bernoulli numbers").
Here the expression [m = 0] has the value 1 if m = 0 and 0 otherwise (Iverson bracket). Whenever a confusion between the two kinds of definitions might arise it can be avoided by referring to the more general definition and by reintroducing the erased parameter: writing B_{m}(0) in the first case and B_{m}(1) in the second will unambiguously denote the value in question.
Explicit definition
Starting again with a slightly more general formula
the choices n = 0 and n = 1 lead to
In 1893 Louis Saalschütz listed a total of 38 explicit formulas for the Bernoulli numbers Template:Harv, usually giving some reference in the older literature.
Generating function
The general formula for the generating function is
The choices n = 0 and n = 1 lead to
Algorithmic description
Although the above recursive formula can be used for computation it is mainly used to establish the connection with the sum of powers because it is computationally expensive. However, both simple and high-end algorithms for computing Bernoulli numbers exist. Pointers to high-end algorithms are given the next section. A simple one is given in pseudocode below.
Input: Integer n≥0. Output: Second Bernoulli number B_{n}.
for m from 0 by 1 to n do A[m] ← 1/(m+1) for j from m by -1 to 1 do A[j-1] ← j×(A[j-1] - A[j]) return A[0] (which is B_{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 Template:Harv which require only O(p (log p)^{2}) operations (see big-O notation).
David Harvey Template:Harv 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 is included in Sage since version 3.1. Prior to that Bernd Kellner Template:Harv computed B_{n} to full precision for n = 10^{6} in December 2002 and Oleksandr Pavlyk Template:Harv 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 1672 3330 G. Fee, S. Plouffe 1996 10000 27677 G. Fee, S. Plouffe 1996 100000 376755 B. C. Kellner 2002 1000000 4767529 O. Pavlyk 2008 10000000 57675260 D. Harvey 2008 100000000 676752569
- Digits is to be understood as the exponent of 10 when B(n) is written as a real in normalized scientific notation.
Different viewpoints and conventions
The Bernoulli numbers can be regarded from four main viewpoints:
- as standalone arithmetical objects,
- as combinatorial objects,
- as values of a sequence of certain polynomials,
- as values of the Riemann zeta function.
Each of these viewpoints leads to a set of more or less different conventions.
- Bernoulli numbers as standalone arithmetical objects.
Associated sequence: 1/6, −1/30, 1/42, −1/30, …
This is the viewpoint of Jakob Bernoulli. (See the cutout from his Ars Conjectandi, first edition, 1713). The Bernoulli numbers are understood as numbers, recursive in nature, invented to solve a certain arithmetical problem, the summation of powers, which is the paradigmatic application of the Bernoulli numbers. These are also the numbers appearing in the Taylor series expansion of tan(x) and tanh(x). It is misleading to call this viewpoint 'archaic'. For example Jean-Pierre Serre uses it in his highly acclaimed book A Course in Arithmetic which is a standard textbook used at many universities today.
- Bernoulli numbers as combinatorial objects.
Associated sequence: 1, +1/2, 1/6, 0, …
This view focuses on the connection between Stirling numbers and Bernoulli numbers and arises naturally in the calculus of finite differences. In its most general and compact form this connection is summarized by the definition of the Stirling polynomials σ_{n}(x), formula (6.52) in Concrete Mathematics by Graham, Knuth and Patashnik.
In consequence B_{n} = n! σ_{n}(1) for n ≥ 0.
- Bernoulli numbers as values of a sequence of certain polynomials.
Assuming the Bernoulli polynomials as already introduced the Bernoulli numbers can be defined in two different ways:
- B_{n} = B_{n}(0). Associated sequence: 1, −1/2, 1/6, 0, …
- B_{n} = B_{n}(1). Associated sequence: 1, +1/2, 1/6, 0, …
The two definitions differ only in the sign of B_{1}. The choice B_{n} = B_{n}(0) is the convention used in the Handbook of Mathematical Functions.
- Bernoulli numbers as values of the Riemann zeta function.
Associated sequence: 1, +1/2, 1/6, 0, …
Using this convention, the values of the Riemann zeta function satisfy nζ(1 − n) = −B_{n} for all integers n≥0. (See the paper of S. C. Woon; the expression nζ(1 − n) for n = 0 is to be understood as lim_{x → 0} xζ(1 − x).)
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 ƒ is a sufficiently often differentiable function the Euler–MacLaurin formula can be written as ^{[4]}
This formulation assumes the convention B_{1} = −1/2. Using the convention B_{1} = 1/2 the formula becomes
Here ƒ^{(0)} = ƒ which is a commonly used notation identifying the zero-th derivative of ƒ with ƒ. Moreover, let ƒ^{(−1)} denote an antiderivative of ƒ. 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 (B_{1} = Template:Frac)
Here denotes the rising factorial power.^{[5]}
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 (again B_{1} = Template:Frac).
Taylor series of tan and tanh
The Bernoulli numbers appear in the Taylor series expansion of the tangent and the hyperbolic tangent functions:
Use in topology
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. 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.
Combinatorial definitions
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 sequence 1, 1/2, 1/3, …
This representation has B_{1} = 1/2.
Worpitzky's representation of the Bernoulli number | ||
B_{0} | = | 1/1 |
B_{1} | = | 1/1 − 1/2 |
B_{2} | = | 1/1 − 3/2 + 2/3 |
B_{3} | = | 1/1 − 7/2 + 12/3 − 6/4 |
B_{4} | = | 1/1 − 15/2 + 50/3 − 60/4 + 24/5 |
B_{5} | = | 1/1 − 31/2 + 180/3 − 390/4 + 360/5 − 120/6 |
B_{6} | = | 1/1 − 63/2 + 602/3 − 2100/4 + 3360/5 − 2520/6 + 720/7 |
A second formula representing the Bernoulli numbers by the Worpitzky numbers is for n ≥ 1
Connection with Stirling numbers of the second kind
If denotes Stirling numbers of the second kind^{[6]} then one has:
where denotes the falling factorial.
If one defines the Bernoulli polynomials as:^{[7]}
where for are the Bernoulli numbers.
Then after the following property of binomial coefficient:
one has,
One also has following for Bernoulli polynomials,^{[7]}
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.^{[8]}^{[9]}^{[10]}
Connection with Stirling numbers of the first kind
The two main formulas relating the unsigned Stirling numbers of the first kind 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 | |||||
n \ m | 0 | 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 A051714/ A051715.
An autosequence is a sequence which has its inverse binomial transform equal to the signed sequence. If the main diagonal is 0's = A000004, the autosequence is of the first kind. Example: A000045, the Fibonacci numbers. If the main diagonal is the first upper diagonal multiplied by 2, it is of the second kind. Example: A164555/ A027642, the second Bernoulli numbers (see A190339). The Akiyama–Tanigawa transform applied to 2^{-n} = 1/ A000079 leads to A198631(n)/ A06519(n+1). Hence:
Akiyama–Tanigawa transform for the second Euler numbers | |||||
n \ m | 0 | 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 A209308 and A227577. A198631(n)/ A006519(n+1) are the second (fractional) Euler numbers and an autosequence of the second kind.
( 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, ... .
Connection with Eulerian numbers
There are formulas connecting Eulerian numbers to Bernoulli numbers:
Both formulas are valid for n ≥ 0 if B_{1} is set to ½. If B_{1} is set to −½ they are valid only for n ≥ 1 and n ≥ 2 respectively.
Connection with Balmer series
A link between Bernoulli numbers and Balmer series could be seen in sequence A191567.
Representation of the second Bernoulli numbers
See A191302. The number are not reduced. Then the columns are easy to find, the denominators being A190339.
Representation of the second Bernoulli numbers | ||
B_{0} | = | 1 = 2/2 |
B_{1} | = | 1/2 |
B_{2} | = | 1/2 − 2/6 |
B_{3} | = | 1/2 − 3/6 |
B_{4} | = | 1/2 − 4/6 + 2/15 |
B_{5} | = | 1/2 − 5/6 + 5/15 |
B_{6} | = | 1/2 − 6/6 + 9/15 − 8/105 |
B_{7} | = | 1/2 − 7/6 + 14/15 − 28/105 |
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 Template:Harv described an algorithm to compute σ_{n}(1) as a binary tree.
Woon's tree for σ_{n}(1) |
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!)).
Asymptotic approximation
The Bernoulli numbers can be expressed in terms of the Riemann zeta function as
It then follows from the Stirling formula that, as n goes to infinity,
Including more terms from the zeta series yields a better approximation, as does factoring in the asymptotic series in Stirling's approximation.
Integral representation and continuation
The integral
has as special values b(2n) = B_{2n} for n > 0.
For example b(3) = (3/2)ζ(3)Π^{−3}Ι and b(5) = −(15/2) ζ(5) Π^{ −5}Ι. Here ζ(n) denotes the Riemann zeta function and Ι the imaginary unit. Already 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 n odd 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 Template:Harv. 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 Template:Harv ‘De summis serierum reciprocarum’ (On the sums of series of reciprocals) and has fascinated mathematicians ever since. The first few of these numbers are
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 ( A132049 and A132050):
These rational numbers also appear in the last paragraph of Euler's paper cited above.
Consider the Akiyama-Tanigawa transform for the sequence A046978(n+2) / A016116(n+1):
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 - A163982/2.
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.
- ( A000111)
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 Template:Harv published an ingenious algorithm which makes it extremely simple to calculate T_{n}.
[begin] 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. [end]
Seidel's algorithm is in fact much more general (see the exposition of Dominique Dumont Template:Harv) and was rediscovered several times thereafter.
Similar to Seidel's approach D. E. Knuth and T. J. Buckholtz Template:Harv 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 Template:Harv 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 A000657, with one 1, and A214267, with two 1's, are in the OEIS.
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 A239005, a signed version of A008280. The main andiagonal is A122045. The main diagonal is A155585. The central column is A099023. Row sums: 1 1 -2 -5 16 61... . See - A163747. See the array beginning with 1 1 0 −2 0 16 0 below.
The Akiyama–Tanigawa algorithm applied to A046978(n + 1) / A016116(n) yields:
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 A122045. Its binomial transform leads to:
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 A155585. The absolute values of the increasing antidiagonals are A008280. The sum of the antidiagonals is − A163747(n + 1).
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 A046978(n) / ( A158780(n + 1) = abs( A117575(n)) + 1 = 1, 2, 2, 3/2, 1, 3/4, 3/4, 7/8, 1, 17/16, 17/16, 33/32... .
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 A000111 could be the numerator of a trigonometric function.
A163747 is an eigensequence of the first kind (the main diagonal is A000004). The corresponding array is:
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) · A002832. The sum of the antidiagonals is 0 −2 0 10... = 2 · A122045(n + 1).
- A163982 is an eigensequence of the second kind, like for instance A164555 / A027642. Hence the array:
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 A099023. The sum of the antidiagonals is 2 0 −4 0... = 2 · A155585(n + 1). Note that A163747 − A163982 = 2 · A122045.
A combinatorial view: alternating permutations
{{#invoke:main|main}}
Around 1880, three years after the publication of Seidel's algorithm, Désiré André proved a now classic result of combinatorial analysis Template:Harv & Template:Harv. 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, A176327 / A027642. Via the second row of its inverse Akiyama–Tanigawa transform A177427, they lead to Balmer series A061037 / A061038.
A companion to the second Bernoulli numbers
See A190339. These numbers are the eigensequence or autosequence of the first kind. 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) = 2 * T (n,k+1) - T(n,k) to T(0,k) = A191754(k)/ A192366(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 A164555/ A027642. For the third row, see A051716.
The first column is 0, 1, 0, -1/3, 0, 7/15, 0, -31/21, 0, 127/105, 0, -511/33, ... from Mersenne numbers, see A141459. For the second column see A140252.
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 Template:Harv, 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 non-zero integers.
Prime numbers with this property are called regular primes. Another classical result of Kummer Template:Harv are the following congruences.
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- 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 , then
Since B_{n} = —n ζ(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 , and so can be extended to a continuous function ζ_{p}(s) for all p-adic integers , 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
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The von Staudt–Clausen theorem was given by Karl Georg Christian von Staudt Template:Harv and Thomas Clausen Template:Harv 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 greater than 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 in this case B_{n} = 0. 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 . 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 Template:Harv 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
denotes the rising factorial power in the notation of D. E. Knuth. The number 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 number.
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 which considered 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.
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 Template:Harv.
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 c-th 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.^{[1]} 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 Template:Harv a rigorous proof of Faulhaber’s formula was first published by Carl Jacobi in 1834 Template:Harv. Donald E. Knuth's in-depth study of Faulhaber's formula concludes:
- “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
- 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 from polynomials in N to polynomials in n.” Template:Harv
Reconstruction of "Summae Potestatum"
The Bernoulli numbers were introduced by Jakob Bernoulli in the book Ars Conjectandi published posthumously in 1713. The main formula can be seen in the second half of the corresponding facsimile. The constant coefficients denoted A, B, C and D by Bernoulli are mapped to the notation which is now prevalent as A = B_{2}, B = B_{4}, C = B_{6}, D = B_{8}. In the expression c·c−1·c−2·c−3 the small dots are used as grouping symbols, not as signs for multiplication. Using today's terminology these expressions are falling factorial powers . The factorial notation k! as a shortcut for 1 × 2 × ... × k was not introduced until 100 years later. 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). (The Mathematics Genealogy Project ^{[11]} shows Leibniz as the doctoral adviser of Jakob Bernoulli. See also the Earliest Uses of Symbols of Calculus.^{[12]}) 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:
In fact this formula imperatively suggests to set B_{1} = ½ when switching from the so-called 'archaic' enumeration which uses only the even indices 2, 4, … to the modern form (more on different conventions in the next paragraph). Most striking in this context is the fact that the falling factorial has for k = 0 the value .^{[13]} Thus Bernoulli's formula can and has to be written:
If B_{1} stands for the value Bernoulli himself has given to the coefficient at that position.
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.^{[14]}
Appendix
Assorted identities
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Values of the first Bernoulli numbers
B_{n} = 0 for all odd n other than 1. For even n, B_{n} is negative if n is divisible by 4 and positive otherwise. The first few non-zero Bernoulli numbers are:
n | Numerator | Denominator | Decimal approximation |
---|---|---|---|
0 | 1 | 1 | +1.00000000000 |
1 | ±1 | 2 | ±0.50000000000 |
2 | 1 | 6 | +0.16666666667 |
4 | −1 | 30 | −0.03333333333 |
6 | 1 | 42 | +0.02380952381 |
8 | −1 | 30 | −0.03333333333 |
10 | 5 | 66 | +0.07575757576 |
12 | −691 | 2730 | −0.25311355311 |
14 | 7 | 6 | +1.16666666667 |
16 | −3617 | 510 | −7.09215686275 |
18 | 43867 | 798 | +54.9711779448 |
A027641 | A027642 |
From 6, the denominators are multiples of the sequence of period 2 : 6,30 A165734. From 2, the denominators are of the form 4*k + 2.
A subsequence of the Bernoulli numbers denominators
A219196 = A027642( A131577) = 1,2,6,30,30,510,510,510,510,131070,131070,131070,131070,131070,131070,131070,131070,8589934590,8589934590,8589934590,8589934590
See also
- Euler number
- Genocchi number
- Kummer's congruences
- poly-Bernoulli number
- Hurwitz zeta function
- Euler summation
- Stirling polynomial
Notes
- ↑ ^{1.0} ^{1.1} Selin, H. (1997), p. 891
- ↑ Smith, D. E. (1914), p. 108
- ↑ Note G in the Menabrea reference
- ↑ Concrete Mathematics, (9.67).
- ↑ Concrete Mathematics, (2.44) and (2.52)
- ↑ L. Comtet, Advanced combinatorics. The art of finite and infinite expansions, Revised and Enlarged Edition, D. Reidel Publ. Co., Dordrecht-Boston, 1974.
- ↑ ^{7.0} ^{7.1} H. Rademacher, Analytic Number Theory, Springer-Verlag, New York, 1973.
- ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑ Mathematics Genealogy Project
- ↑ Earliest Uses of Symbols of Calculus
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- ↑ Template:Harvnb
References
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External links
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- 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.
- Template:Cite web
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