Английская Википедия:Bernoulli number
Шаблон:Short description Шаблон:Use American English Шаблон:Use shortened footnotes
| Шаблон:Mvar | fraction | decimal |
|---|---|---|
| 0 | 1 | +1.000000000 |
| 1 | ±Шаблон:Sfrac | ±0.500000000 |
| 2 | Шаблон:Sfrac | +0.166666666 |
| 3 | 0 | +0.000000000 |
| 4 | −Шаблон:Sfrac | −0.033333333 |
| 5 | 0 | +0.000000000 |
| 6 | Шаблон:Sfrac | +0.023809523 |
| 7 | 0 | +0.000000000 |
| 8 | −Шаблон:Sfrac | −0.033333333 |
| 9 | 0 | +0.000000000 |
| 10 | Шаблон:Sfrac | +0.075757575 |
| 11 | 0 | +0.000000000 |
| 12 | −Шаблон:Sfrac | −0.253113553 |
| 13 | 0 | +0.000000000 |
| 14 | Шаблон:Sfrac | +1.166666666 |
| 15 | 0 | +0.000000000 |
| 16 | −Шаблон:Sfrac | −7.092156862 |
| 17 | 0 | +0.000000000 |
| 18 | Шаблон:Sfrac | +54.97117794 |
| 19 | 0 | +0.000000000 |
| 20 | −Шаблон:Sfrac | −529.1242424 |
In mathematics, the Bernoulli numbers Шаблон:Math are a sequence of rational numbers which occur frequently in analysis. 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.
The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by <math>B^{-{}}_n</math> and <math>B^{+{}}_n</math>; they differ only for Шаблон:Math, where <math>B^{-{}}_1=-1/2</math> and <math>B^{+{}}_1=+1/2</math>. For every odd Шаблон:Math, Шаблон:Math. For every even Шаблон:Math, Шаблон:Math is negative if Шаблон:Math is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoulli polynomials <math>B_n(x)</math>, with <math>B^{-{}}_n=B_n(0)</math> and <math>B^+_n=B_n(1)</math>.Шаблон:R
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 Takakazu. Seki's discovery was posthumously published in 1712Шаблон:R in his work Katsuyō Sanpō; Bernoulli's, also posthumously, in his Ars Conjectandi of 1713. Ada Lovelace's note G on the Analytical Engine from 1842 describes an algorithm prepared by Babbage for generating Bernoulli numbers with Babbage's machine.Шаблон:R As a result, the Bernoulli numbers have the distinction of being the subject of the first published complex computer program.
Notation
The superscript Шаблон:Math used in this article distinguishes the two sign conventions for Bernoulli numbers. Only the Шаблон:Math term is affected:
- Шаблон:Math with Шаблон:Math (Шаблон:OEIS2C / Шаблон:OEIS2C) is the sign convention prescribed by NIST and most modern textbooks.Шаблон:Sfnp
- Шаблон:Math with Шаблон:Math (Шаблон:OEIS2C / Шаблон:OEIS2C) was used in the older literature,Шаблон:R and (since 2022) by Donald Knuth[1] following Peter Luschny's "Bernoulli Manifesto".[2]
In the formulas below, one can switch from one sign convention to the other with the relation <math>B_n^{+}=(-1)^n B_n^{-}</math>, or for integer Шаблон:Mvar = 2 or greater, simply ignore it.
Since Шаблон:Math for all odd Шаблон:Math, and many formulas only involve even-index Bernoulli numbers, a few authors write "Шаблон:Math" instead of Шаблон:Math. This article does not follow that notation.
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 Шаблон:Mvar positive integers, the sum of the squares and of the cubes of the first Шаблон:Mvar 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 Шаблон:Mathth powers of the first Шаблон:Math positive integers for Шаблон:Math.
The Swiss mathematician Jakob Bernoulli (1654–1705) was the first to realize the existence of a single sequence of constants Шаблон:Math which provide a uniform formula for all sums of powers.Шаблон:Sfnp
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 Шаблон:Mvarth powers for any positive integer Шаблон:Math 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 Takakazu independently discovered the Bernoulli numbers and his result was published a year earlier, also posthumously, in 1712.Шаблон:R 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. According to KnuthШаблон:Sfnp a rigorous proof of Faulhaber's formula was first published by Carl Jacobi in 1834.Шаблон:R 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 Шаблон:Math ... would provide a uniform
- <math display=inline>\sum n^m = \frac 1{m+1}\left( B_0n^{m+1}-\binom{m+1} 1 B_1 n^m+\binom{m+1} 2B_2n^{m-1}-\cdots +(-1)^m\binom{m+1}mB_mn\right) </math>
- 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 Шаблон:Math from polynomials in Шаблон:Mvar to polynomials in Шаблон:Mvar."Шаблон:Sfnp
In the above Knuth meant <math>B_1^-</math>; instead using <math>B_1^+</math> the formula avoids subtraction:
- <math display=inline> \sum n^m = \frac 1{m+1}\left( B_0n^{m+1}+\binom{m+1} 1 B^+_1 n^m+\binom{m+1} 2B_2n^{m-1}+\cdots+\binom{m+1}mB_mn\right). </math>
Reconstruction of "Summae Potestatum"
The Bernoulli numbers Шаблон:OEIS2C(n)/Шаблон:OEIS2C(n) were introduced by Jakob Bernoulli in the book Ars Conjectandi published posthumously in 1713 page 97. The main formula can be seen in the second half of the corresponding facsimile. The constant coefficients denoted Шаблон:Math, Шаблон:Math, Шаблон:Math and Шаблон:Math by Bernoulli are mapped to the notation which is now prevalent as Шаблон:Math, Шаблон:Math, Шаблон:Math, Шаблон:Math. The expression Шаблон:Math means Шаблон:Math – the small dots are used as grouping symbols. Using today's terminology these expressions are falling factorial powers Шаблон:Math. The factorial notation Шаблон:Math as a shortcut for Шаблон:Math 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 Шаблон:Math for "summa" (sum).Шаблон:Efn The letter Шаблон:Math 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 Шаблон:Math. Putting things together, for positive Шаблон:Math, today a mathematician is likely to write Bernoulli's formula as:
- <math> \sum_{k=1}^n k^c = \frac{n^{c+1}}{c+1}+\frac 1 2 n^c+\sum_{k=2}^c \frac{B_k}{k!} c^{\underline{k-1}}n^{c-k+1}.</math>
This formula suggests setting Шаблон:Math 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 Шаблон:Math has for Шаблон:Math the value Шаблон:Math.Шаблон:Sfnp Thus Bernoulli's formula can be written
- <math> \sum_{k=1}^n k^c = \sum_{k=0}^c \frac{B_k}{k!}c^{\underline{k-1}} n^{c-k+1}</math>
if Шаблон:Math, recapturing the value Bernoulli gave to the coefficient at that position.
The formula for <math>\textstyle \sum_{k=1}^n k^9</math> in the first half of the quotation by Bernoulli above contains an error at the last term; it should be <math>-\tfrac {3}{20}n^2</math> instead of <math>-\tfrac {1}{12}n^2</math>.
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 integral expression.
For the proof of the equivalence of the four approaches.[3]
Recursive definition
The Bernoulli numbers obey the sum formulasШаблон:R
- <math> \begin{align} \sum_{k=0}^{m}\binom {m+1} k B^{-{}}_k &= \delta_{m, 0} \\ \sum_{k=0}^{m}\binom {m+1} k B^{+{}}_k &= m+1 \end{align}</math>
where <math>m=0,1,2...</math> and Шаблон:Math denotes the Kronecker delta. Solving for <math>B^{\mp{}}_m</math> gives the recursive formulas
- <math>\begin{align}
B_m^{-{}} &= \delta_{m, 0} - \sum_{k=0}^{m-1} \binom{m}{k} \frac{B^{-{}}_k}{m - k + 1} \\
B_m^+ &= 1 - \sum_{k=0}^{m-1} \binom{m}{k} \frac{B^+_k}{m - k + 1}.
\end{align}</math>
Explicit definition
In 1893 Louis Saalschütz listed a total of 38 explicit formulas for the Bernoulli numbers,Шаблон:R usually giving some reference in the older literature. One of them is (for <math>m\geq 1</math>):
- <math>\begin{align}
B^{-{}}_m &= \sum_{k=0}^m \sum_{v=0}^k (-1)^v \binom{k}{v} \frac{v^m}{k + 1} \\
B^+_m &= \sum_{k=0}^m \sum_{v=0}^k (-1)^v \binom{k}{v} \frac{(v + 1)^m}{k + 1}.
\end{align}</math>
Generating function
The exponential generating functions are
- <math>\begin{alignat}{3}
\frac{t}{e^t - 1} &= \frac{t}{2} \left( \operatorname{coth} \frac{t}{2} -1 \right) &&= \sum_{m=0}^\infty \frac{B^{-{}}_m t^m}{m!}\\
\frac{t}{1 - e^{-t}} &= \frac{t}{2} \left( \operatorname{coth} \frac{t}{2} +1 \right) &&= \sum_{m=0}^\infty \frac{B^+_m t^m}{m!}.
\end{alignat}</math> where the substitution is <math>t \to - t</math>. If we let <math>F(t)=\sum_{i=1}^\infty f_it^i</math> and <math>G(t)=1/(1+F(t))=\sum_{i=0}^\infty g_it^i</math> then
- <math>G(t)=1-F(t)G(t).</math>
Then <math>g_0=1</math> and for <math>m>0</math> the mШаблон:Sup term in the series for <math>G(t)</math> is:
- <math>g_mt^i=-\sum_{j=0}^{m-1}f_{m-j}g_jt^m</math>
If
- <math>F(t)=\frac{e^t-1}t-1=\sum_{i=1}^\infty \frac{t^i}{(i+1)!}</math>
then we find that
- <math>G(t)=t/(e^t-1)</math>
- <math>\begin{align}
m!g_m&=-\sum_{j=0}^{m-1}\frac{m!}{j!}\frac{j!g_j}{(m-j+1)!}\\ &=-\frac 1{m+1}\sum_{j=0}^{m-1}\binom{m+1}jj!g_j\\ \end{align}</math>
showing that the values of <math>i!g_i</math> obey the recursive formula for the Bernoulli numbers <math>B^-_i</math>.
The (ordinary) generating function
- <math> z^{-1} \psi_1(z^{-1}) = \sum_{m=0}^{\infty} B^+_m z^m</math>
is an asymptotic series. It contains the trigamma function Шаблон:Math.
Integral Expression
From the generating functions above, one can obtain the following integral formula for the even Bernoulli numbers:
- <math>B_{2n} = 4n (-1)^{n+1} \int_0^{\infty} \frac{t^{2n-1}}{e^{2 \pi t} -1 } \mathrm{d} t </math>
Bernoulli numbers and the Riemann zeta function
The Bernoulli numbers can be expressed in terms of the Riemann zeta function:
- Шаблон:Math for Шаблон:Math .
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:Шаблон:Sfnp
- <math> B_{2n} = \frac {(-1)^{n+1}2(2n)!} {(2\pi)^{2n}} \zeta(2n) \quad </math> for Шаблон:Math .
Now the argument of the zeta function is positive.
It then follows from Шаблон:Math (Шаблон:Math) and Stirling's formula that
- <math> |B_{2 n}| \sim 4 \sqrt{\pi n} \left(\frac{n}{ \pi e} \right)^{2n} \quad </math> for Шаблон:Math .
Efficient computation of Bernoulli numbers
In some applications it is useful to be able to compute the Bernoulli numbers Шаблон:Math through Шаблон:Math modulo Шаблон:Mvar, where Шаблон:Mvar is a prime; for example to test whether Vandiver's conjecture holds for Шаблон:Mvar, or even just to determine whether Шаблон:Mvar 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) Шаблон:Math arithmetic operations would be required. Fortunately, faster methods have been developedШаблон:R which require only Шаблон:Math operations (see [[big-O notation|big Шаблон:Mvar notation]]).
David HarveyШаблон:R describes an algorithm for computing Bernoulli numbers by computing Шаблон:Math modulo Шаблон:Mvar for many small primes Шаблон:Mvar, and then reconstructing Шаблон:Math via the Chinese remainder theorem. Harvey writes that the asymptotic time complexity of this algorithm is Шаблон:Math and claims that this implementation is significantly faster than implementations based on other methods. Using this implementation Harvey computed Шаблон:Math for Шаблон:Math. Harvey's implementation has been included in SageMath since version 3.1. Prior to that, Bernd KellnerШаблон:R computed Шаблон:Math to full precision for Шаблон:Math in December 2002 and Oleksandr PavlykШаблон:R for Шаблон:Math 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 Шаблон:Val Шаблон:Val G. Fee, S. Plouffe 1996 Шаблон:Val Шаблон:Val G. Fee, S. Plouffe 1996 Шаблон:Val Шаблон:Val B. C. Kellner 2002 Шаблон:Val Шаблон:Val O. Pavlyk 2008 Шаблон:Val Шаблон:Val D. Harvey 2008 Шаблон:Val Шаблон:Val
- * Digits is to be understood as the exponent of 10 when Шаблон:Math is written as a real number in normalized scientific notation.
A possible algorithm for computing Bernoulli numbers in the Julia programming language is given byШаблон:R
b = Array{Float64}(undef, n+1)
b[1] = 1
b[2] = -0.5
for m=2:n
for k=0:m
for v=0:k
b[m+1] += (-1)^v * binomial(k,v) * v^(m) / (k+1)
end
end
end
return b
Applications of the Bernoulli numbers
Asymptotic analysis
Arguably the most important application of the Bernoulli numbers in mathematics is their use in the Euler–Maclaurin formula. Assuming that Шаблон:Mvar is a sufficiently often differentiable function the Euler–Maclaurin formula can be written asШаблон:Sfnp
- <math>\sum_{k=a}^{b-1} f(k) = \int_a^b f(x)\,dx + \sum_{k=1}^m \frac{B^-_k}{k!} (f^{(k-1)}(b)-f^{(k-1)}(a))+R_-(f,m).</math>
This formulation assumes the convention Шаблон:Math. Using the convention Шаблон:Math the formula becomes
- <math>\sum_{k=a+1}^{b} f(k) = \int_a^b f(x)\,dx + \sum_{k=1}^m \frac{B^+_k}{k!} (f^{(k-1)}(b)-f^{(k-1)}(a))+R_+(f,m).</math>
Here <math>f^{(0)}=f</math> (i.e. the zeroth-order derivative of <math>f</math> is just <math>f</math>). Moreover, let <math>f^{(-1)}</math> denote an antiderivative of <math>f</math>. By the fundamental theorem of calculus,
- <math>\int_a^b f(x)\,dx = f^{(-1)}(b) - f^{(-1)}(a).</math>
Thus the last formula can be further simplified to the following succinct form of the Euler–Maclaurin formula
- <math> \sum_{k=a+1}^{b} f(k)= \sum_{k=0}^m \frac{B_k}{k!} (f^{(k-1)}(b)-f^{(k-1)}(a))+R(f,m). </math>
This form is for example the source for the important Euler–Maclaurin expansion of the zeta function
- <math> \begin{align}
\zeta(s) & =\sum_{k=0}^m \frac{B^+_k}{k!} s^{\overline{k-1}} + R(s,m) \\
& = \frac{B_0}{0!}s^{\overline{-1}} + \frac{B^+_1}{1!} s^{\overline{0}} + \frac{B_2}{2!} s^{\overline{1}} +\cdots+R(s,m) \\
& = \frac{1}{s-1} + \frac{1}{2} + \frac{1}{12}s + \cdots + R(s,m).
\end{align} </math>
Here Шаблон:Math denotes the rising factorial power.Шаблон:Sfnp
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 Шаблон:Math.
- <math>\psi(z) \sim \ln z - \sum_{k=1}^\infty \frac{B^+_k}{k z^k} </math>
Sum of powers
Шаблон:Main Bernoulli numbers feature prominently in the closed form expression of the sum of the Шаблон:Mathth powers of the first Шаблон:Math positive integers. For Шаблон:Math define
- <math>S_m(n) = \sum_{k=1}^n k^m = 1^m + 2^m + \cdots + n^m. </math>
This expression can always be rewritten as a polynomial in Шаблон:Math of degree Шаблон:Math. The coefficients of these polynomials are related to the Bernoulli numbers by Bernoulli's formula:
- <math>S_m(n) = \frac{1}{m + 1} \sum_{k=0}^m \binom{m + 1}{k} B^+_k n^{m + 1 - k} = m! \sum_{k=0}^m \frac{B^+_k n^{m + 1 - k}}{k! (m+1-k)!} ,</math>
where Шаблон:Math denotes the binomial coefficient.
For example, taking Шаблон:Math to be 1 gives the triangular numbers Шаблон:Math Шаблон:OEIS2C.
- <math> 1 + 2 + \cdots + n = \frac{1}{2} (B_0 n^2 + 2 B^+_1 n^1) = \tfrac12 (n^2 + n).</math>
Taking Шаблон:Math to be 2 gives the square pyramidal numbers Шаблон:Math Шаблон:OEIS2C.
- <math>1^2 + 2^2 + \cdots + n^2 = \frac{1}{3} (B_0 n^3 + 3 B^+_1 n^2 + 3 B_2 n^1) = \tfrac13 \left(n^3 + \tfrac32 n^2 + \tfrac12 n\right).</math>
Some authors use the alternate convention for Bernoulli numbers and state Bernoulli's formula in this way:
- <math>S_m(n) = \frac{1}{m + 1} \sum_{k=0}^m (-1)^k \binom{m + 1}{k} B^{-{}}_k n^{m + 1 - k}.</math>
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|Шаблон:Mvar-analog]].Шаблон:R
Taylor series
The Bernoulli numbers appear in the Taylor series expansion of many trigonometric functions and hyperbolic functions.
<math display="block">\begin{align} \tan x &= \hphantomШаблон:1\over x \sum_{n=1}^\infty \frac{(-1)^{n-1} 2^{2n} (2^{2n}-1) B_{2n} }{(2n)!}\; x^{2n-1}, && \left|x \right| < \frac \pi 2. \\ \cot x &= {1\over x} \sum_{n=0}^\infty \frac{(-1)^n B_{2n} (2x)^{2n}}{(2n)!}, & 0 < & |x| < \pi. \\ \tanh x &= \hphantomШаблон:1\over x \sum_{n=1}^\infty \frac{2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}\;x^{2n-1}, && |x| < \frac \pi 2. \\ \coth x &= {1\over x} \sum_{n=0}^\infty \frac{B_{2n} (2x)^{2n}}{(2n)!}, & 0 < & |x| < \pi. \end{align}</math>
Laurent series
The Bernoulli numbers appear in the following Laurent series:Шаблон:Sfnp
Digamma function: <math> \psi(z)= \ln z- \sum_{k=1}^\infty \frac {B_k^{+{}}} {k z^k} </math>
Use in topology
The Kervaire–Milnor formula for the order of the cyclic group of diffeomorphism classes of [[exotic sphere|exotic Шаблон:Math-spheres]] which bound parallelizable manifolds involves Bernoulli numbers. Let Шаблон:Math be the number of such exotic spheres for Шаблон:Math, then
- <math>\textit{ES}_n = (2^{2n-2}-2^{4n-3}) \operatorname{Numerator}\left(\frac{B_{4n}}{4n} \right) .</math>
The Hirzebruch signature theorem for the [[Hirzebruch signature theorem#L genus and the Hirzebruch signature theorem|Шаблон:Mvar 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 Шаблон:Math and the power function Шаблон:Math is employed. The signless Worpitzky numbers are defined as
- <math> W_{n,k}=\sum_{v=0}^k (-1)^{v+k} (v+1)^n \frac{k!}{v!(k-v)!} . </math>
They can also be expressed through the Stirling numbers of the second kind
- <math> W_{n,k}=k! \left\{ {n+1\atop k+1} \right\}.</math>
A Bernoulli number is then introduced as an inclusion–exclusion sum of Worpitzky numbers weighted by the harmonic sequence 1, Шаблон:Sfrac, Шаблон:Sfrac, ...
- <math> B_{n}=\sum_{k=0}^n (-1)^k \frac{W_{n,k}}{k+1}\ =\ \sum_{k=0}^n \frac{1}{k+1} \sum_{v=0}^k (-1)^v (v+1)^n {k \choose v}\ . </math>
This representation has Шаблон:Math.
Consider the sequence Шаблон:Math, Шаблон:Math. From Worpitzky's numbers Шаблон:OEIS2C, Шаблон:OEIS2C applied to Шаблон:Math is identical to the Akiyama–Tanigawa transform applied to Шаблон:Math (see Connection with Stirling numbers of the first kind). This can be seen via the table:
Identity of
Worpitzky's representation and Akiyama–Tanigawa transform1 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 Шаблон:Math.
Hence for the second fractional Euler numbers Шаблон:OEIS2C (Шаблон:Math) / Шаблон:OEIS2C (Шаблон:Math):
A second formula representing the Bernoulli numbers by the Worpitzky numbers is for Шаблон:Math
- <math> B_n=\frac n {2^{n+1}-2}\sum_{k=0}^{n-1} (-2)^{-k}\, W_{n-1,k} . </math>
The simplified second Worpitzky's representation of the second Bernoulli numbers is:
Шаблон:OEIS2C (Шаблон:Math) / Шаблон:OEIS2C(Шаблон:Math) = Шаблон:Math × Шаблон:OEIS2C(Шаблон:Math) / Шаблон:OEIS2C(Шаблон:Math)
which links the second Bernoulli numbers to the second fractional Euler numbers. The beginning is:
The numerators of the first parentheses are Шаблон:OEIS2C (see Connection with Stirling numbers of the first kind).
Connection with Stirling numbers of the second kind
If one defines the Bernoulli polynomials Шаблон:Math as:Шаблон:R
- <math> B_k(j)=k\sum_{m=0}^{k-1}\binom{j}{m+1}S(k-1,m)m!+B_k </math>
where Шаблон:Math for Шаблон:Math are the Bernoulli numbers.
One also has the following for Bernoulli polynomials,Шаблон:R
- <math> B_k(j)=\sum_{n=0}^k \binom{k}{n} B_n j^{k-n}. </math>
The coefficient of Шаблон:Mvar in Шаблон:Math is Шаблон:Math.
Comparing the coefficient of Шаблон:Mvar in the two expressions of Bernoulli polynomials, one has:
- <math> B_k=\sum_{m=0}^{k-1} (-1)^m \frac{m!}{m+1} S(k-1,m)</math>
(resulting in Шаблон:Math) which is an explicit formula for Bernoulli numbers and can be used to prove Von-Staudt Clausen theorem.Шаблон:R
Connection with Stirling numbers of the first kind
The two main formulas relating the unsigned Stirling numbers of the first kind Шаблон:Math to the Bernoulli numbers (with Шаблон:Math) are
- <math> \frac{1}{m!}\sum_{k=0}^m (-1)^{k} \left[{m+1\atop k+1}\right] B_k = \frac{1}{m+1}, </math>
and the inversion of this sum (for Шаблон:Math, Шаблон:Math)
- <math> \frac{1}{m!}\sum_{k=0}^m (-1)^k \left[{m+1\atop k+1}\right] B_{n+k} = A_{n,m}. </math>
Here the number Шаблон:Math are the rational Akiyama–Tanigawa numbers, the first few of which are displayed in the following table.
Akiyama–Tanigawa number Шаблон:Diagonal split header 0 1 2 3 4 0 1 Шаблон:Sfrac Шаблон:Sfrac Шаблон:Sfrac Шаблон:Sfrac 1 Шаблон:Sfrac Шаблон:Sfrac Шаблон:Sfrac Шаблон:Sfrac ... 2 Шаблон:Sfrac Шаблон:Sfrac Шаблон:Sfrac ... ... 3 0 Шаблон:Sfrac ... ... ... 4 −Шаблон:Sfrac ... ... ... ...
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 Шаблон:OEIS2C/Шаблон:OEIS2C.
An autosequence is a sequence which has its inverse binomial transform equal to the signed sequence. If the main diagonal is zeroes = Шаблон:OEIS2C, the autosequence is of the first kind. Example: Шаблон:OEIS2C, the Fibonacci numbers. If the main diagonal is the first upper diagonal multiplied by 2, it is of the second kind. Example: Шаблон:OEIS2C/Шаблон:OEIS2C, the second Bernoulli numbers (see Шаблон:OEIS2C). The Akiyama–Tanigawa transform applied to Шаблон:Math = 1/Шаблон:OEIS2C leads to Шаблон:OEIS2C (n) / Шаблон:OEIS2C (n + 1). Hence:
Akiyama–Tanigawa transform for the second Euler numbers Шаблон:Diagonal split header 0 1 2 3 4 0 1 Шаблон:Sfrac Шаблон:Sfrac Шаблон:Sfrac Шаблон:Sfrac 1 Шаблон:Sfrac Шаблон:Sfrac Шаблон:Sfrac Шаблон:Sfrac ... 2 0 Шаблон:Sfrac Шаблон:Sfrac ... ... 3 −Шаблон:Sfrac −Шаблон:Sfrac ... ... ... 4 0 ... ... ... ...
See Шаблон:OEIS2C and Шаблон:OEIS2C. Шаблон:OEIS2C (Шаблон:Math) / Шаблон:OEIS2C (Шаблон:Math) are the second (fractional) Euler numbers and an autosequence of the second kind.
- (Шаблон:Sfrac = Шаблон:Math) × ( Шаблон:Math = Шаблон:Math) = Шаблон:Sfrac = Шаблон:Math.
Also valuable for Шаблон:OEIS2C / Шаблон:OEIS2C (see Connection with Worpitzky numbers).
Connection with Pascal's triangle
There are formulas connecting Pascal's triangle to Bernoulli numbersШаблон:Efn
- <math> B^{+}_n=\frac{|A_n|}{(n+1)!}EducationBot (обсуждение)</math>
where <math>|A_n|</math> is the determinant of a n-by-n Hessenberg matrix part of Pascal's triangle whose elements are: <math> a_{i, k} = \begin{cases} 0 & \text{if } k>1+i \\ {i+1 \choose k-1} & \text{otherwise} \end{cases} </math>
Example:
- <math> B^{+}_6 =\frac{\det\begin{pmatrix}
1& 2& 0& 0& 0& 0\\ 1& 3& 3& 0& 0& 0\\ 1& 4& 6& 4& 0& 0\\ 1& 5& 10& 10& 5& 0\\ 1& 6& 15& 20& 15& 6\\ 1& 7& 21& 35& 35& 21 \end{pmatrix}}{7!}=\frac{120}{5040}=\frac 1 {42} </math>
Connection with Eulerian numbers
There are formulas connecting Eulerian numbers Шаблон:Math to Bernoulli numbers:
- <math>\begin{align}
\sum_{m=0}^n (-1)^m \left \langle {n\atop m} \right \rangle &= 2^{n+1} (2^{n+1}-1) \frac{B_{n+1}}{n+1}, \\ \sum_{m=0}^n (-1)^m \left \langle {n\atop m} \right \rangle \binom{n}{m}^{-1} &= (n+1) B_n. \end{align}</math>
Both formulae are valid for Шаблон:Math if Шаблон:Math is set to Шаблон:Sfrac. If Шаблон:Math is set to −Шаблон:Sfrac they are valid only for Шаблон:Math and Шаблон:Math respectively.
A binary tree representation
The Stirling polynomials Шаблон:Math are related to the Bernoulli numbers by Шаблон:Math. S. C. Woon described an algorithm to compute Шаблон:Math as a binary tree:Шаблон:R
Woon's recursive algorithm (for Шаблон:Math) starts by assigning to the root node Шаблон:Math. Given a node Шаблон:Math of the tree, the left child of the node is Шаблон:Math and the right child Шаблон:Math. A node Шаблон:Math is written as Шаблон:Math in the initial part of the tree represented above with ± denoting the sign of Шаблон:Math.
Given a node Шаблон:Mvar the factorial of Шаблон:Mvar is defined as
- <math> N! = a_1 \prod_{k=2}^{\operatorname{length}(N)} a_k!. </math>
Restricted to the nodes Шаблон:Mvar of a fixed tree-level Шаблон:Mvar the sum of Шаблон:Math is Шаблон:Math, thus
- <math> B_n = \sum_\stackrel{N \text{ node of}}{\text{ tree-level } n} \frac{n!}{N!}. </math>
For example:
Integral representation and continuation
The integral
- <math> b(s) = 2e^{s i \pi/2}\int_0^\infty \frac{st^s}{1-e^{2\pi t}} \frac{dt}{t} = \frac{s!}{2^{s-1}}\frac{\zeta(s)}{{ }\pi^s{ }}(-i)^s= \frac{2s!\zeta(s)}{(2\pi i)^s}</math>
has as special values Шаблон:Math for Шаблон:Math.
For example, Шаблон:Math and Шаблон:Math. Here, Шаблон:Mvar is the Riemann zeta function, and Шаблон:Mvar is the imaginary unit. Leonhard Euler (Opera Omnia, Ser. 1, Vol. 10, p. 351) considered these numbers and calculated
- <math> \begin{align}
p &= \frac{3}{2\pi^3}\left(1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots \right) = 0.0581522\ldots \\
q &= \frac{15}{2\pi^5}\left(1+\frac{1}{2^5}+\frac{1}{3^5}+\cdots \right) = 0.0254132\ldots
\end{align}</math>
Another similar integral representation is
- <math> b(s) = -\frac{e^{s i \pi/2}}{2^{s}-1}\int_0^\infty \frac{st^{s}}{\sinh\pi t} \frac{dt}{t}= \frac{2e^{s i \pi/2}}{2^{s}-1}\int_0^\infty \frac{e^{\pi t}st^s}{1-e^{2\pi t}} \frac{dt}{t}. </math>
The relation to the Euler numbers and Шаблон:Pi
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 Шаблон:Math are in magnitude approximately Шаблон:Math times larger than the Bernoulli numbers Шаблон:Math. In consequence:
- <math> \pi \sim 2 (2^{2n} - 4^{2n}) \frac{B_{2n}}{E_{2n}}. </math>
This asymptotic equation reveals that Шаблон:Pi lies in the common root of both the Bernoulli and the Euler numbers. In fact Шаблон:Pi could be computed from these rational approximations.
Bernoulli numbers can be expressed through the Euler numbers and vice versa. Since, for odd Шаблон:Mvar, Шаблон:Math (with the exception Шаблон:Math), it suffices to consider the case when Шаблон:Mvar is even.
- <math>\begin{align}
B_n &= \sum_{k=0}^{n-1}\binom{n-1}{k} \frac{n}{4^n-2^n}E_k & n&=2, 4, 6, \ldots \\[6pt]
E_n &= \sum_{k=1}^n \binom{n}{k-1} \frac{2^k-4^k}{k} B_k & n&=2,4,6,\ldots
\end{align}</math>
These conversion formulas express a connection 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 Шаблон:Pi. These numbers are defined for Шаблон:Math as[4]Шаблон:R
- <math> S_n = 2 \left(\frac{2}{\pi}\right)^n \sum_{k = 0}^\infty \frac{ (-1)^{kn} }{(2k+1)^n} = 2 \left(\frac{2}{\pi}\right)^n \lim_{K\to \infty} \sum_{k = -K}^K (4k+1)^{-n}. </math>
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 De summis serierum reciprocarum (On the sums of series of reciprocals) and has fascinated mathematicians ever since.Шаблон:R The first few of these numbers are
- <math> S_n = 1,1,\frac{1}{2},\frac{1}{3},\frac{5}{24}, \frac{2}{15},\frac{61}{720},\frac{17}{315},\frac{277}{8064},\frac{62}{2835},\ldots </math> (Шаблон:OEIS2C / Шаблон:OEIS2C)
These are the coefficients in the expansion of Шаблон:Math.
The Bernoulli numbers and Euler numbers can be understood as special views of these numbers, selected from the sequence Шаблон:Math and scaled for use in special applications.
- <math>\begin{align}
B_{n} &= (-1)^{\left\lfloor \frac{n}{2}\right\rfloor} [ n \text{ even}] \frac{n! }{2^n - 4^n}\, S_{n}\ , & n&= 2, 3, \ldots \\
E_n &= (-1)^{\left\lfloor \frac{n}{2}\right\rfloor} [ n \text{ even}] n! \, S_{n+1} & n &= 0, 1, \ldots
\end{align}</math>
The expression [[[:Шаблон:Math]] even] has the value 1 if Шаблон:Math 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 Шаблон:Math when Шаблон:Mvar is even. The Шаблон:Math are rational approximations to Шаблон:Pi and two successive terms always enclose the true value of Шаблон:Pi. Beginning with Шаблон:Math the sequence starts (Шаблон:OEIS2C / Шаблон:OEIS2C):
- <math> 2, 4, 3, \frac{16}{5}, \frac{25}{8}, \frac{192}{61}, \frac{427}{136}, \frac{4352}{1385}, \frac{12465}{3968}, \frac{158720}{50521},\ldots \quad \longrightarrow \pi. </math>
These rational numbers also appear in the last paragraph of Euler's paper cited above.
Consider the Akiyama–Tanigawa transform for the sequence Шаблон:OEIS2C (Шаблон:Math) / Шаблон:OEIS2C (Шаблон:Math):
From the second, the numerators of the first column are the denominators of Euler's formula. The first column is −Шаблон:Sfrac × Шаблон:OEIS2C.
An algorithmic view: the Seidel triangle
The sequence Sn has another unexpected yet important property: The denominators of Sn+1 divide the factorial Шаблон:Math. In other words: the numbers Шаблон:Math, sometimes called Euler zigzag numbers, are integers.
- <math> T_n = 1,\,1,\,1,\,2,\,5,\,16,\,61,\,272,\,1385,\,7936,\,50521,\,353792,\ldots \quad n=0, 1, 2, 3, \ldots </math> (Шаблон:OEIS2C). See (Шаблон:OEIS2C).
Their exponential generating function is the sum of the secant and tangent functions.
- <math> \sum_{n=0}^\infty T_n \frac{x^n}{n!} = \tan \left(\frac\pi4 + \frac x2\right) = \sec x + \tan x</math>.
Thus the above representations of the Bernoulli and Euler numbers can be rewritten in terms of this sequence as
- <math>\begin{align}
B_n &= (-1)^{\left\lfloor \frac{n}{2}\right\rfloor} [n\text{ even}] \frac{n }{2^n-4^n}\, T_{n-1}\ & n &\geq 2 \\
E_n &= (-1)^{\left\lfloor \frac{n}{2}\right\rfloor} [n\text{ even}] T_{n} & n &\geq 0
\end{align}</math>
These identities make it easy to compute the Bernoulli and Euler numbers: the Euler numbers Шаблон:Math are given immediately by Шаблон:Math and the Bernoulli numbers Шаблон:Math are fractions obtained from Шаблон:Math by some easy shifting, avoiding rational arithmetic.
What remains is to find a convenient way to compute the numbers Шаблон:Math. However, already in 1877 Philipp Ludwig von Seidel published an ingenious algorithm, which makes it simple to calculate Шаблон:Math.Шаблон:R
- Start by putting 1 in row 0 and let Шаблон:Math denote the number of the row currently being filled
- If Шаблон:Math is odd, then put the number on the left end of the row Шаблон:Math in the first position of the row Шаблон:Math, 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 Шаблон:Math is even, proceed similar in the other direction.
Seidel's algorithm is in fact much more general (see the exposition of Dominique Dumont Шаблон:R) and was rediscovered several times thereafter.
Similar to Seidel's approach D. E. Knuth and T. J. Buckholtz gave a recurrence equation for the numbers Шаблон:Math and recommended this method for computing Шаблон:Math and Шаблон:Math 'on electronic computers using only simple operations on integers'.Шаблон:R
V. I. ArnoldШаблон:R rediscovered Seidel's algorithm 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 Шаблон:OEIS2C, with one 1, and Шаблон:OEIS2C, with two 1s, 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 Шаблон:OEIS2C, a signed version of Шаблон:OEIS2C. The main andiagonal is Шаблон:OEIS2C. The main diagonal is Шаблон:OEIS2C. The central column is Шаблон:OEIS2C. Row sums: 1, 1, −2, −5, 16, 61.... See Шаблон:OEIS2C. See the array beginning with 1, 1, 0, −2, 0, 16, 0 below.
The Akiyama–Tanigawa algorithm applied to Шаблон:OEIS2C (Шаблон:Math) / Шаблон:OEIS2C(Шаблон:Math) yields:
1 1 Шаблон:Sfrac 0 −Шаблон:Sfrac −Шаблон:Sfrac −Шаблон:Sfrac 0 1 Шаблон:Sfrac 1 0 −Шаблон:Sfrac −1 −1 Шаблон:Sfrac 4 Шаблон:Sfrac 0 −5 −Шаблон:Sfrac 1 5 5 −Шаблон:Sfrac 0 61 −61
1. The first column is Шаблон:OEIS2C. 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 Шаблон:OEIS2C. The absolute values of the increasing antidiagonals are Шаблон:OEIS2C. The sum of the antidiagonals is Шаблон:Nowrap
2. The second column is Шаблон:Nowrap. 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 Шаблон:Nowrap. 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 Шаблон:OEIS2C (Шаблон:Math) / (Шаблон:OEIS2C (Шаблон:Math) = abs(Шаблон:OEIS2C (Шаблон:Mvar)) + 1 = Шаблон:Nowrap.
1 2 2 Шаблон:Sfrac 1 Шаблон:Sfrac Шаблон:Sfrac −1 0 Шаблон:Sfrac 2 Шаблон:Sfrac 0 −1 −3 −Шаблон:Sfrac 3 Шаблон:Sfrac 2 −3 −Шаблон:Sfrac −13 5 21 −Шаблон:Sfrac −16 45 −61
The first column whose the absolute values are Шаблон:OEIS2C could be the numerator of a trigonometric function.
Шаблон:OEIS2C is an autosequence of the first kind (the main diagonal is Шаблон:OEIS2C). 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 Шаблон:Nowrap = Шаблон:Math × Шаблон:OEIS2C. The sum of the antidiagonals is Шаблон:Nowrap = 2 × Шаблон:OEIS2C(n + 1).
−Шаблон:OEIS2C is an autosequence of the second kind, like for instance Шаблон:OEIS2C / Шаблон:OEIS2C. 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 Шаблон:Nowrap, is the double of the first upper one, here Шаблон:OEIS2C. The sum of the antidiagonals is Шаблон:Nowrap = 2 × Шаблон:OEIS2C(Шаблон:Math1). Шаблон:OEIS2C − Шаблон:OEIS2C = 2 × Шаблон:OEIS2C.
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.Шаблон:R Looking at the first terms of the Taylor expansion of the trigonometric functions Шаблон:Math and Шаблон:Math André made a startling discovery.
- <math>\begin{align}
\tan x &= x + \frac{2x^3}{3!} + \frac{16x^5}{5!} + \frac{272x^7}{7!} + \frac{7936x^9}{9!} + \cdots\\[6pt]
\sec x &= 1 + \frac{x^2}{2!} + \frac{5x^4}{4!} + \frac{61x^6}{6!} + \frac{1385x^8}{8!} + \frac{50521x^{10}}{10!} + \cdots
\end{align}</math>
The coefficients are the Euler numbers of odd and even index, respectively. In consequence the ordinary expansion of Шаблон:Math has as coefficients the rational numbers Шаблон:Math.
- <math> \tan x + \sec x = 1 + x + \tfrac{1}{2}x^2 + \tfrac{1}{3}x^3 + \tfrac{5}{24}x^4 + \tfrac{2}{15}x^5 + \tfrac{61}{720}x^6 + \cdots </math>
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: Шаблон:Math, Шаблон:Math, Шаблон:Math, Шаблон:Math, Шаблон:Math, Шаблон:OEIS2C / Шаблон:OEIS2C. Via the second row of its inverse Akiyama–Tanigawa transform Шаблон:OEIS2C, they lead to Balmer series Шаблон:OEIS2C / Шаблон:OEIS2C.
The Akiyama–Tanigawa algorithm applied to Шаблон:OEIS2C (Шаблон:Math) / Шаблон:OEIS2C (Шаблон:Mvar) leads to the Bernoulli numbers Шаблон:OEIS2C / Шаблон:OEIS2C, Шаблон:OEIS2C / Шаблон:OEIS2C, or Шаблон:OEIS2C Шаблон:OEIS2C without Шаблон:Math, named intrinsic Bernoulli numbers Шаблон:Math.
Hence another link between the intrinsic Bernoulli numbers and the Balmer series via Шаблон:OEIS2C (Шаблон:Math).
Шаблон:OEIS2C (Шаблон:Math) = 0, 2, 1, 6,... is a permutation of the non-negative numbers.
The terms of the first row are f(n) = Шаблон:Math. 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, g(n), is an autosequence of the second kind.
Euler Шаблон:OEIS2C (Шаблон:Math) / Шаблон:OEIS2C (Шаблон:Math) without the second term (Шаблон:Sfrac) are the fractional intrinsic Euler numbers Шаблон:Math The corresponding Akiyama transform is:
The first line is Шаблон:Math. Шаблон:Math 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 Шаблон:OEIS2C preceded by 0. The difference table is:
Arithmetical properties of the Bernoulli numbers
The Bernoulli numbers can be expressed in terms of the Riemann zeta function as Шаблон:Math for integers Шаблон:Math provided for Шаблон:Math the expression Шаблон:Math is understood as the limiting value and the convention Шаблон:Math 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 Шаблон:Mvar is a prime number if and only if Шаблон:Math is congruent to −1 modulo Шаблон:Mvar. 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,Шаблон:R which says:
- If the odd prime Шаблон:Mvar does not divide any of the numerators of the Bernoulli numbers Шаблон:Math then Шаблон:Math has no solutions in nonzero integers.
Prime numbers with this property are called regular primes. Another classical result of Kummer are the following congruences.Шаблон:R
- Let Шаблон:Mvar be an odd prime and Шаблон:Mvar an even number such that Шаблон:Math does not divide Шаблон:Mvar. Then for any non-negative integer Шаблон:Mvar
- <math> \frac{B_{k(p-1)+b}}{k(p-1)+b} \equiv \frac{B_{b}}{b} \pmod{p}. </math>
A generalization of these congruences goes by the name of Шаблон:Math-adic continuity.
Шаблон:Math-adic continuity
If Шаблон:Mvar, Шаблон:Mvar and Шаблон:Mvar are positive integers such that Шаблон:Mvar and Шаблон:Mvar are not divisible by Шаблон:Math and Шаблон:Math, then
- <math>(1-p^{m-1})\frac{B_m}{m} \equiv (1-p^{n-1})\frac{B_n} n \pmod{p^b}.</math>
Since Шаблон:Math, this can also be written
- <math>\left(1-p^{-u}\right)\zeta(u) \equiv \left(1-p^{-v}\right)\zeta(v) \pmod{p^b},</math>
where Шаблон:Math and Шаблон:Math, so that Шаблон:Mvar and Шаблон:Mvar are nonpositive and not congruent to 1 modulo Шаблон:Math. This tells us that the Riemann zeta function, with Шаблон:Math taken out of the Euler product formula, is continuous in the [[p-adic number|Шаблон:Mvar-adic number]]s on odd negative integers congruent modulo Шаблон:Math to a particular Шаблон:Math, and so can be extended to a continuous function Шаблон:Math for all Шаблон:Mvar-adic integers <math>\mathbb{Z}_p,</math> the [[p-adic zeta function|Шаблон:Mvar-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:
- <math>\binom{m+3}{m} B_m=\begin{cases}
\frac{m+3}{3}-\sum\limits_{j=1}^\frac{m}{6}\binom{m+3}{m-6j}B_{m-6j}, & \text{if } m\equiv 0\pmod 6;\\ \frac{m+3}{3}-\sum\limits_{j=1}^\frac{m-2}{6}\binom{m+3}{m-6j}B_{m-6j}, & \text{if } m\equiv 2\pmod 6;\\ -\frac{m+3}{6}-\sum\limits_{j=1}^\frac{m-4}{6}\binom{m+3}{m-6j}B_{m-6j}, & \text{if } m\equiv 4\pmod 6.\end{cases}</math>
Von Staudt–Clausen theorem
Шаблон:Main The von Staudt–Clausen theorem was given by Karl Georg Christian von StaudtШаблон:R and Thomas ClausenШаблон:R independently in 1840. The theorem states that for every Шаблон:Math,
- <math> B_{2n} + \sum_{(p-1)\,\mid\,2n} \frac1p</math>
is an integer. The sum extends over all primes Шаблон:Math for which Шаблон:Math divides Шаблон:Math.
A consequence of this is that the denominator of Шаблон:Math is given by the product of all primes Шаблон:Math for which Шаблон:Math divides Шаблон:Math. In particular, these denominators are square-free and divisible by 6.
Why do the odd Bernoulli numbers vanish?
The sum
- <math>\varphi_k(n) = \sum_{i=0}^n i^k - \frac{n^k} 2</math>
can be evaluated for negative values of the index Шаблон:Math. Doing so will show that it is an odd function for even values of Шаблон:Math, which implies that the sum has only terms of odd index. This and the formula for the Bernoulli sum imply that Шаблон:Math is 0 for Шаблон:Math even and Шаблон:Math; and that the term for Шаблон:Math 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 Шаблон:Math the number Шаблон:Math 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
- <math> 2B_n =\sum_{m=0}^n (-1)^m \frac{2}{m+1}m! \left\{{n+1\atop m+1} \right\} = 0\quad(n>1 \text{ is odd})</math>
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 Шаблон:Math be the number of surjective maps from Шаблон:Math} to Шаблон:Math}, then Шаблон:Math. The last equation can only hold if
- <math> \sum_{\text{odd }m=1}^{n-1} \frac 2 {m^2}S_{n,m}=\sum_{\text{even } m=2}^n \frac{2}{m^2} S_{n,m} \quad (n>2 \text{ is even}). </math>
This equation can be proved by induction. The first two examples of this equation are
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 numbers. In fact Marcel Riesz proved that the RH is equivalent to the following assertion:Шаблон:R
- For every Шаблон:Math there exists a constant Шаблон:Math (depending on Шаблон:Math) such that Шаблон:Math as Шаблон:Math.
Here Шаблон:Math is the Riesz function
- <math> R(x) = 2 \sum_{k=1}^\infty
\frac{k^{\overline{k}} x^{k}}{(2\pi)^{2k}\left(\frac{B_{2k}}{2k}\right)} = 2\sum_{k=1}^\infty \frac{k^{\overline{k}}x^k}{(2\pi)^{2k}\beta_{2k}}. </math>
Шаблон:Math denotes the rising factorial power in the notation of D. E. Knuth. The numbers Шаблон:Math occur frequently in the study of the zeta function and are significant because Шаблон:Math is a Шаблон:Math-integer for primes Шаблон:Math where Шаблон:Math does not divide Шаблон:Math. The Шаблон:Math 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-function|Dirichlet Шаблон:Mvar-functions]] in the same way that Bernoulli numbers are related to special values of the Riemann zeta function.
Let Шаблон:Mvar be a Dirichlet character modulo Шаблон:Mvar. The generalized Bernoulli numbers attached to Шаблон:Mvar are defined by
- <math>\sum_{a=1}^f \chi(a) \frac{te^{at}}{e^{ft}-1} = \sum_{k=0}^\infty B_{k,\chi}\frac{t^k}{k!}.</math>
Apart from the exceptional Шаблон:Math, we have, for any Dirichlet character Шаблон:Mvar, that Шаблон:Math if Шаблон:Math.
Generalizing the relation between Bernoulli numbers and values of the Riemann zeta function at non-positive integers, one has the for all integers Шаблон:Math:
- <math>L(1-k,\chi)=-\frac{B_{k,\chi}}k,</math>
where Шаблон:Math is the Dirichlet Шаблон:Mvar-function of Шаблон:Mvar.Шаблон:R
Eisenstein–Kronecker number
Шаблон:Main Eisenstein–Kronecker numbers are an analogue of the generalized Bernoulli numbers for imaginary quadratic fields.Шаблон:R They are related to critical L-values of Hecke characters.Шаблон:R
Appendix
Assorted identities
Шаблон:Unordered list{n-k}\frac{B_k}{k} - \sum_{k=2}^{n-2} \binom{n}{k}\frac{B_{n-k}}{n-k} B_k =H_n B_n</math>
|11 = Let Шаблон:Math. Yuri Matiyasevich found (1997)
- <math> (n+2)\sum_{k=2}^{n-2}B_k B_{n-k}-2\sum_{l=2}^{n-2}\binom{n+2}{l} B_l B_{n-l}=n(n+1)B_n </math>
|12 = Faber–Pandharipande–Zagier–Gessel identity: for Шаблон:Math,
- <math> \frac{n}{2}\left(B_{n-1}(x)+\sum_{k=1}^{n-1}\frac{B_{k}(x)}{k}
\frac{B_{n-k}(x)}{n-k}\right) -\sum_{k=0}^{n-1}\binom{n}{k}\frac{B_{n-k}} {n-k} B_k(x) =H_{n-1}B_n(x).</math>
Choosing Шаблон:Math or Шаблон:Math results in the Bernoulli number identity in one or another convention.
|13 = The next formula is true for Шаблон:Math if Шаблон:Math, but only for Шаблон:Math if Шаблон:Math.
- <math> \sum_{k=0}^n \binom{n}{k} \frac{B_k}{n-k+2} = \frac{B_{n+1}}{n+1} </math>
|14 = Let Шаблон:Math. Then
- <math> -1 + \sum_{k=0}^n \binom{n}{k} \frac{2^{n-k+1}}{n-k+1}B_k(1) = 2^n </math>
and
- <math> -1 + \sum_{k=0}^n \binom{n}{k} \frac{2^{n-k+1}}{n-k+1}B_{k}(0) = \delta_{n,0} </math>
|15 = A reciprocity relation of M. B. Gelfand:Шаблон:R
- <math> (-1)^{m+1} \sum_{j=0}^k \binom{k}{j} \frac{B_{m+1+j}}{m+1+j} + (-1)^{k+1} \sum_{j=0}^m \binom{m}{j}\frac{B_{k+1+j}}{k+1+j} = \frac{k!m!}{(k+m+1)!} </math>
}}
See also
- Bernoulli polynomial
- Bernoulli polynomials of the second kind
- Bernoulli umbra
- Bell number
- Euler number
- Genocchi number
- Kummer's congruences
- Poly-Bernoulli number
- Hurwitz zeta function
- Euler summation
- Stirling polynomial
- Sums of powers
Notes
References
Bibliography
- Шаблон:Citation.
- Шаблон:Citation
- Шаблон:Citation.
- Шаблон:Citation.
- Шаблон:Citation.
- Шаблон:Citation.
- Шаблон:Citation.
- Шаблон:Citation.
- Шаблон:Cite arXiv.
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation.
- Шаблон:Citation.
- Шаблон:Citation
- Шаблон:Citation.
- Шаблон:Citation.
- Шаблон:Citation.
- Шаблон:Citation.
- Шаблон:Citation.
- Шаблон:Citation.
- Шаблон:Citation.
- Шаблон:Citation.
- Шаблон:Citation.
- Шаблон:Cite arXiv.
- Шаблон:Citation.
External links
- Шаблон:SpringerEOM
- The first 498 Bernoulli Numbers from Project Gutenberg
- A multimodular algorithm for computing Bernoulli numbers
- The Bernoulli Number Page
- Bernoulli number programs at LiteratePrograms
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
Шаблон:Calculus topics Шаблон:Authority control
- ↑ Donald Knuth (2022), Recent News (2022): Concrete Mathematics and Bernoulli. Шаблон:Blockquote
- ↑ Peter Luschny (2013), The Bernoulli Manifesto
- ↑ See Шаблон:Harvp or Шаблон:Harvp.
- ↑ Шаблон:Citation
