Template:Short description Template:Use American English Template:Use shortened footnotes
| Template:Mvar | fraction | decimal |
|---|---|---|
| 0 | 1 | +1.000000000 |
| 1 | ±Template:Sfrac | ±0.500000000 |
| 2 | Template:Sfrac | +0.166666666 |
| 3 | 0 | +0.000000000 |
| 4 | −Template:Sfrac | −0.033333333 |
| 5 | 0 | +0.000000000 |
| 6 | Template:Sfrac | +0.023809523 |
| 7 | 0 | +0.000000000 |
| 8 | −Template:Sfrac | −0.033333333 |
| 9 | 0 | +0.000000000 |
| 10 | Template:Sfrac | +0.075757575 |
| 11 | 0 | +0.000000000 |
| 12 | −Template:Sfrac | −0.253113553 |
| 13 | 0 | +0.000000000 |
| 14 | Template:Sfrac | +1.166666666 |
| 15 | 0 | +0.000000000 |
| 16 | −Template:Sfrac | −7.092156862 |
| 17 | 0 | +0.000000000 |
| 18 | Template:Sfrac | +54.97117794 |
| 19 | 0 | +0.000000000 |
| 20 | −Template:Sfrac | −529.1242424 |
In mathematics, the Bernoulli numbers Template: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 Template:Math, where <math>B^{-{}}_1=-1/2</math> and <math>B^{+{}}_1=+1/2</math>. For every odd Template:Math, Template:Math. For every even Template:Math, Template:Math is negative if Template: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>.Template: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 1712Template: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 for generating Bernoulli numbers with Babbage's machine;Template:R it is disputed whether Lovelace or Babbage developed the algorithm. As a result, the Bernoulli numbers have the distinction of being the subject of the first published complex computer program.
NotationEdit
The superscript Template:Math used in this article distinguishes the two sign conventions for Bernoulli numbers. Only the Template:Math term is affected:
- Template:Math with Template:Math (Template:OEIS2C / Template:OEIS2C) is the sign convention prescribed by NIST and most modern textbooks.Template:Sfnp
- Template:Math with Template:Math (Template:OEIS2C / Template:OEIS2C) was used in the older literature,Template:R and (since 2022) by Donald Knuth<ref>Donald Knuth (2022), Recent News (2022): Concrete Mathematics and Bernoulli. <templatestyles src="Template:Blockquote/styles.css" />
But last year I took a close look at Peter Luschny's Bernoulli manifesto, where he gives more than a dozen good reasons why the value of $B_1$ should really be plus one-half. He explains that some mathematicians of the early 20th century had unilaterally changed the conventions, because some of their formulas came out a bit nicer when the negative value was used. It was their well-intentioned but ultimately poor choice that had led to what I'd been taught in the 1950s. […] By now, hundreds of books that use the “minus-one-half” convention have unfortunately been written. Even worse, all the major software systems for symbolic mathematics have that 20th-century aberration deeply embedded. Yet Luschny convinced me that we have all been wrong, and that it's high time to change back to the correct definition before the situation gets even worse.
{{#if:|{{#if:|}}
}}{{#invoke:Check for unknown parameters|check|unknown=Template:Main other|preview=Page using Template:Blockquote with unknown parameter "_VALUE_"|ignoreblank=y| 1 | 2 | 3 | 4 | 5 | author | by | char | character | cite | class | content | multiline | personquoted | publication | quote | quotesource | quotetext | sign | source | style | text | title | ts }}</ref> following Peter Luschny's "Bernoulli Manifesto".<ref>Peter Luschny (2013), The Bernoulli Manifesto</ref>
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 Template:Mvar = 2 or greater, simply ignore it.
Since Template:Math for all odd Template:Math, and many formulas only involve even-index Bernoulli numbers, a few authors write "Template:Math" instead of Template:Math. This article does not follow that notation.
HistoryEdit
Early historyEdit
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 Template:Mvar positive integers, the sum of the squares and of the cubes of the first Template: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 Template:Math to the sums of the Template:Mathth powers of the first Template:Math positive integers for Template:Math.
The Swiss mathematician Jakob Bernoulli (1654–1705) was the first to realize the existence of a single sequence of constants Template:Math which provide a uniform formula for all sums of powers.Template: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 Template:Mvarth powers for any positive integer Template: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.Template: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 KnuthTemplate:Sfnp a rigorous proof of Faulhaber's formula was first published by Carl Jacobi in 1834.Template: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 Template: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 Template:Math from polynomials in Template:Mvar to polynomials in Template:Mvar."Template: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"Edit
The Bernoulli numbers Template:OEIS2C(n)/Template: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 Template:Math, Template:Math, Template:Math and Template:Math by Bernoulli are mapped to the notation which is now prevalent as Template:Math, Template:Math, Template:Math, Template:Math. The expression Template:Math means Template:Math – the small dots are used as grouping symbols. Using today's terminology these expressions are falling factorial powers Template:Math. The factorial notation Template:Math as a shortcut for Template: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 Template:Math for "summa" (sum).Template:Efn The letter Template: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 Template:Math. Putting things together, for positive Template: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 Template: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 Template:Math has for Template:Math the value Template:Math.Template: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 Template: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>.
DefinitionsEdit
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, see Template:Harvp or Template:Harvp.
Recursive definitionEdit
The Bernoulli numbers obey the sum formulasTemplate: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 Template:Math denotes the Kronecker delta.
The first of these is sometimes written<ref>Jordan (1950) p 233</ref> as the formula (for m > 1) <math display=block>(B+1)^m-B_m=0,</math> where the power is expanded formally using the binomial theorem and <math>B^k</math> is replaced by <math>B_k</math>.
Solving for <math>B^{\mp{}}_m</math> gives the recursive formulas<ref>Ireland and Rosen (1990) p 229</ref>
- <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 definitionEdit
In 1893 Louis Saalschütz listed a total of 38 explicit formulas for the Bernoulli numbers,Template: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 \frac1{k+1} \sum_{j=0}^k \binom{k}{j} (-1)^j j^m \\
B^+_m &= \sum_{k=0}^m \frac1{k+1} \sum_{j=0}^k \binom{k}{j} (-1)^j (j + 1)^m.
\end{align}</math>
Generating functionEdit
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{te^t}{e^t - 1} = \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>. The two generating functions only differ by t. Template:Collapse top 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 mth 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>. Template:Collapse bottom
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 Template:Math.
Integral ExpressionEdit
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 functionEdit
The Bernoulli numbers can be expressed in terms of the Riemann zeta function:
- Template:Math for Template:Math .
Here the argument of the zeta function is 0 or negative. As <math>\zeta(k)</math> is zero for negative even integers (the trivial zeroes), if n>1 is odd, <math>\zeta(1-n)</math> is zero.
By means of the zeta functional equation and the gamma reflection formula the following relation can be obtained:Template:Sfnp
- <math> B_{2n} = \frac {(-1)^{n+1}2(2n)!} {(2\pi)^{2n}} \zeta(2n) \quad </math> for Template:Math .
Now the argument of the zeta function is positive.
It then follows from Template:Math (Template: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 Template:Math .
Efficient computation of Bernoulli numbersEdit
In some applications it is useful to be able to compute the Bernoulli numbers Template:Math through Template:Math modulo Template:Mvar, where Template:Mvar is a prime; for example to test whether Vandiver's conjecture holds for Template:Mvar, or even just to determine whether Template: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) Template:Math arithmetic operations would be required. Fortunately, faster methods have been developedTemplate:R which require only Template:Math operations (see [[big-O notation|big Template:Mvar notation]]).
David HarveyTemplate:R describes an algorithm for computing Bernoulli numbers by computing Template:Math modulo Template:Mvar for many small primes Template:Mvar, and then reconstructing Template:Math via the Chinese remainder theorem. Harvey writes that the asymptotic time complexity of this algorithm is Template:Math and claims that this implementation is significantly faster than implementations based on other methods. Using this implementation Harvey computed Template:Math for Template:Math. Harvey's implementation has been included in SageMath since version 3.1. Prior to that, Bernd KellnerTemplate:R computed Template:Math to full precision for Template:Math in December 2002 and Oleksandr PavlykTemplate:R for Template: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 Template:Val Template:Val G. Fee, S. Plouffe 1996 Template:Val Template:Val G. Fee, S. Plouffe 1996 Template:Val Template:Val B. C. Kellner 2002 Template:Val Template:Val O. Pavlyk 2008 Template:Val Template:Val D. Harvey 2008 Template:Val Template:Val
- * Digits is to be understood as the exponent of 10 when Template:Math is written as a real number in normalized scientific notation.
Applications of the Bernoulli numbersEdit
Asymptotic analysisEdit
Arguably the most important application of the Bernoulli numbers in mathematics is their use in the Euler–Maclaurin formula. Assuming that Template:Mvar is a sufficiently often differentiable function the Euler–Maclaurin formula can be written asTemplate: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 Template:Math. Using the convention Template: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 Template:Math denotes the rising factorial power.Template: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 Template:Math.
- <math>\psi(z) \sim \ln z - \sum_{k=1}^\infty \frac{B^+_k}{k z^k} </math>
Sum of powersEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Bernoulli numbers feature prominently in the closed form expression of the sum of the Template:Mathth powers of the first Template:Math positive integers. For Template: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 Template:Math of degree Template: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 Template:Math denotes the binomial coefficient.
For example, taking Template:Math to be 1 gives the triangular numbers Template:Math Template:OEIS2C.
- <math> 1 + 2 + \cdots + n = \frac{1}{2} (B_0 n^2 + 2 B^+_1 n^1) = \tfrac12 (n^2 + n).</math>
Taking Template:Math to be 2 gives the square pyramidal numbers Template:Math Template: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|Template:Mvar-analog]].Template:R
Taylor seriesEdit
The Bernoulli numbers appear in the Taylor series expansion of many trigonometric functions and hyperbolic functions.
<math display="block">\begin{align} \tan x &= \hphantomTemplate: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 &= \hphantomTemplate: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 seriesEdit
The Bernoulli numbers appear in the following Laurent series:Template:Sfnp
Digamma function: <math> \psi(z)= \ln z- \sum_{k=1}^\infty \frac {B_k^{+{}}} {k z^k} </math>
Use in topologyEdit
The Kervaire–Milnor formula for the order of the cyclic group of diffeomorphism classes of [[exotic sphere|exotic Template:Math-spheres]] which bound parallelizable manifolds involves Bernoulli numbers. Let Template:Math be the number of such exotic spheres for Template: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|Template:Mvar genus]] of a smooth oriented closed manifold of dimension 4n also involves Bernoulli numbers.
Connections with combinatorial numbersEdit
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 numbersEdit
The definition to proceed with was developed by Julius Worpitzky in 1883. Besides elementary arithmetic only the factorial function Template:Math and the power function Template: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, Template:Sfrac, Template: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 Template:Math.
Consider the sequence Template:Math, Template:Math. From Worpitzky's numbers Template:OEIS2C, Template:OEIS2C applied to Template:Math is identical to the Akiyama–Tanigawa transform applied to Template: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 Template:Math.
Hence for the second fractional Euler numbers Template:OEIS2C (Template:Math) / Template:OEIS2C (Template:Math):
A second formula representing the Bernoulli numbers by the Worpitzky numbers is for Template: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:
Template:OEIS2C (Template:Math) / Template:OEIS2C(Template:Math) = Template:Math × Template:OEIS2C(Template:Math) / Template:OEIS2C(Template:Math)
which links the second Bernoulli numbers to the second fractional Euler numbers. The beginning is:
The numerators of the first parentheses are Template:OEIS2C (see Connection with Stirling numbers of the first kind).
Connection with Stirling numbers of the second kindEdit
If one defines the Bernoulli polynomials Template:Math as:Template:R
- <math> B_k(j)=k\sum_{m=0}^{k-1}\binom{j}{m+1}S(k-1,m)m!+B_k </math>
where Template:Math for Template:Math are the Bernoulli numbers, and Template:Math is a Stirling number of the second kind.
One also has the following for Bernoulli polynomials,Template:R
- <math> B_k(j)=\sum_{n=0}^k \binom{k}{n} B_n j^{k-n}. </math>
The coefficient of Template:Mvar in Template:Math is Template:Math.
Comparing the coefficient of Template: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 Template:Math) which is an explicit formula for Bernoulli numbers and can be used to prove Von-Staudt Clausen theorem.Template:R
Connection with Stirling numbers of the first kindEdit
The two main formulas relating the unsigned Stirling numbers of the first kind Template:Math to the Bernoulli numbers (with Template: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 Template:Math, Template: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 Template:Math are the rational Akiyama–Tanigawa numbers, the first few of which are displayed in the following table.
Akiyama–Tanigawa number Template:Diagonal split header 0 1 2 3 4 0 1 Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac 1 Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac ... 2 Template:Sfrac Template:Sfrac Template:Sfrac ... ... 3 0 Template:Sfrac ... ... ... 4 −Template: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 Template:OEIS2C/Template:OEIS2C.
An autosequence is a sequence which has its inverse binomial transform equal to the signed sequence. If the main diagonal is zeroes = Template:OEIS2C, the autosequence is of the first kind. Example: Template:OEIS2C, the Fibonacci numbers. If the main diagonal is the first upper diagonal multiplied by 2, it is of the second kind. Example: Template:OEIS2C/Template:OEIS2C, the second Bernoulli numbers (see Template:OEIS2C). The Akiyama–Tanigawa transform applied to Template:Math = 1/Template:OEIS2C leads to Template:OEIS2C (n) / Template:OEIS2C (n + 1). Hence:
Akiyama–Tanigawa transform for the second Euler numbers Template:Diagonal split header 0 1 2 3 4 0 1 Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac 1 Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac ... 2 0 Template:Sfrac Template:Sfrac ... ... 3 −Template:Sfrac −Template:Sfrac ... ... ... 4 0 ... ... ... ...
See Template:OEIS2C and Template:OEIS2C. Template:OEIS2C (Template:Math) / Template:OEIS2C (Template:Math) are the second (fractional) Euler numbers and an autosequence of the second kind.
- (Template:Sfrac = Template:Math) × ( Template:Math = Template:Math) = Template:Sfrac = Template:Math.
Also valuable for Template:OEIS2C / Template:OEIS2C (see Connection with Worpitzky numbers).
Connection with Pascal's triangleEdit
There are formulas connecting Pascal's triangle to Bernoulli numbersTemplate:Efn
- <math> B^{+}_n=\frac{|A_n|}{(n+1)!}~~~</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 numbersEdit
There are formulas connecting Eulerian numbers Template: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 Template:Math if Template:Math is set to Template:Sfrac. If Template:Math is set to −Template:Sfrac they are valid only for Template:Math and Template:Math respectively.
A binary tree representationEdit
The Stirling polynomials Template:Math are related to the Bernoulli numbers by Template:Math. S. C. Woon described an algorithm to compute Template:Math as a binary tree:Template:R
Woon's recursive algorithm (for Template:Math) starts by assigning to the root node Template:Math. Given a node Template:Math of the tree, the left child of the node is Template:Math and the right child Template:Math. A node Template:Math is written as Template:Math in the initial part of the tree represented above with ± denoting the sign of Template:Math.
Given a node Template:Mvar the factorial of Template:Mvar is defined as
- <math> N! = a_1 \prod_{k=2}^{\operatorname{length}(N)} a_k!. </math>
Restricted to the nodes Template:Mvar of a fixed tree-level Template:Mvar the sum of Template:Math is Template:Math, thus
- <math> B_n = \sum_\stackrel{N \text{ node of}}{\text{ tree-level } n} \frac{n!}{N!}. </math>
For example:
Integral representation and continuationEdit
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 Template:Math for Template:Math.
For example, Template:Math and Template:Math. Here, Template:Mvar is the Riemann zeta function, and Template: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 Template:PiEdit
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 Template:Math are in magnitude approximately Template:Math times larger than the Bernoulli numbers Template:Math. In consequence:
- <math> \pi \sim 2 (2^{2n} - 4^{2n}) \frac{B_{2n}}{E_{2n}}. </math>
This asymptotic equation reveals that Template:Pi lies in the common root of both the Bernoulli and the Euler numbers. In fact Template:Pi could be computed from these rational approximations.
Bernoulli numbers can be expressed through the Euler numbers and vice versa. Since, for odd Template:Mvar, Template:Math (with the exception Template:Math), it suffices to consider the case when Template: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 Template:Pi. These numbers are defined for Template:Math as<ref>Template:Citation</ref>Template: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.Template: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> (Template:OEIS2C / Template:OEIS2C)
These are the coefficients in the expansion of Template:Math.
The Bernoulli numbers and Euler numbers can be understood as special views of these numbers, selected from the sequence Template: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 [[[:Template:Math]] even] has the value 1 if Template: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 Template:Math when Template:Mvar is even. The Template:Math are rational approximations to Template:Pi and two successive terms always enclose the true value of Template:Pi. Beginning with Template:Math the sequence starts (Template:OEIS2C / Template: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 Template:OEIS2C (Template:Math) / Template:OEIS2C (Template:Math):
From the second, the numerators of the first column are the denominators of Euler's formula. The first column is −Template:Sfrac × Template:OEIS2C.
An algorithmic view: the Seidel triangleEdit
The sequence Sn has another unexpected yet important property: The denominators of Sn+1 divide the factorial Template:Math. In other words: the numbers Template: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> (Template:OEIS2C). See (Template: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 Template:Math are given immediately by Template:Math and the Bernoulli numbers Template:Math are fractions obtained from Template:Math by some easy shifting, avoiding rational arithmetic.
What remains is to find a convenient way to compute the numbers Template:Math. However, already in 1877 Philipp Ludwig von Seidel published an ingenious algorithm, which makes it simple to calculate Template:Math.Template:R
- Start by putting 1 in row 0 and let Template:Math denote the number of the row currently being filled
- If Template:Math is odd, then put the number on the left end of the row Template:Math in the first position of the row Template: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 Template:Math is even, proceed similar in the other direction.
Seidel's algorithm is in fact much more general (see the exposition of Dominique Dumont Template: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 Template:Math and recommended this method for computing Template:Math and Template:Math 'on electronic computers using only simple operations on integers'.Template:R
V. I. ArnoldTemplate: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 Template:OEIS2C, with one 1, and Template: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 Template:OEIS2C, a signed version of Template:OEIS2C. The main andiagonal is Template:OEIS2C. The main diagonal is Template:OEIS2C. The central column is Template:OEIS2C. Row sums: 1, 1, −2, −5, 16, 61.... See Template:OEIS2C. See the array beginning with 1, 1, 0, −2, 0, 16, 0 below.
The Akiyama–Tanigawa algorithm applied to Template:OEIS2C (Template:Math) / Template:OEIS2C(Template:Math) yields:
1 1 Template:Sfrac 0 −Template:Sfrac −Template:Sfrac −Template:Sfrac 0 1 Template:Sfrac 1 0 −Template:Sfrac −1 −1 Template:Sfrac 4 Template:Sfrac 0 −5 −Template:Sfrac 1 5 5 −Template:Sfrac 0 61 −61
1. The first column is Template: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 Template:OEIS2C. The absolute values of the increasing antidiagonals are Template:OEIS2C. The sum of the antidiagonals is Template:Nowrap
2. The second column is Template: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 Template: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 Template:OEIS2C (Template:Math) / (Template:OEIS2C (Template:Math) = abs(Template:OEIS2C (Template:Mvar)) + 1 = Template:Nowrap.
1 2 2 Template:Sfrac 1 Template:Sfrac Template:Sfrac −1 0 Template:Sfrac 2 Template:Sfrac 0 −1 −3 −Template:Sfrac 3 Template:Sfrac 2 −3 −Template:Sfrac −13 5 21 −Template:Sfrac −16 45 −61
The first column whose the absolute values are Template:OEIS2C could be the numerator of a trigonometric function.
Template:OEIS2C is an autosequence of the first kind (the main diagonal is Template: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 Template:Nowrap = Template:Math × Template:OEIS2C. The sum of the antidiagonals is Template:Nowrap = 2 × Template:OEIS2C(n + 1).
−Template:OEIS2C is an autosequence of the second kind, like for instance Template:OEIS2C / Template: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 Template:Nowrap, is the double of the first upper one, here Template:OEIS2C. The sum of the antidiagonals is Template:Nowrap = 2 × Template:OEIS2C(Template:Math1). Template:OEIS2C − Template:OEIS2C = 2 × Template:OEIS2C.
A combinatorial view: alternating permutationsEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}
Around 1880, three years after the publication of Seidel's algorithm, Désiré André proved a now classic result of combinatorial analysis.Template:R Looking at the first terms of the Taylor expansion of the trigonometric functions Template:Math and Template: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 Template:Math has as coefficients the rational numbers Template: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 sequencesEdit
The arithmetic mean of the first and the second Bernoulli numbers are the associate Bernoulli numbers: Template:Math, Template:Math, Template:Math, Template:Math, Template:Math, Template:OEIS2C / Template:OEIS2C. Via the second row of its inverse Akiyama–Tanigawa transform Template:OEIS2C, they lead to Balmer series Template:OEIS2C / Template:OEIS2C.
The Akiyama–Tanigawa algorithm applied to Template:OEIS2C (Template:Math) / Template:OEIS2C (Template:Mvar) leads to the Bernoulli numbers Template:OEIS2C / Template:OEIS2C, Template:OEIS2C / Template:OEIS2C, or Template:OEIS2C Template:OEIS2C without Template:Math, named intrinsic Bernoulli numbers Template:Math.
Hence another link between the intrinsic Bernoulli numbers and the Balmer series via Template:OEIS2C (Template:Math).
Template:OEIS2C (Template:Math) = 0, 2, 1, 6,... is a permutation of the non-negative numbers.
The terms of the first row are f(n) = Template: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 Template:OEIS2C (Template:Math) / Template:OEIS2C (Template:Math) without the second term (Template:Sfrac) are the fractional intrinsic Euler numbers Template:Math The corresponding Akiyama transform is:
The first line is Template:Math. Template: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 Template:OEIS2C preceded by 0. The difference table is:
Arithmetical properties of the Bernoulli numbersEdit
The Bernoulli numbers can be expressed in terms of the Riemann zeta function as Template:Math for integers Template:Math provided for Template:Math the expression Template:Math is understood as the limiting value and the convention Template: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 Template:Mvar is a prime number if and only if Template:Math is congruent to −1 modulo Template: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 theoremsEdit
The Bernoulli numbers are related to Fermat's Last Theorem (FLT) by Kummer's theorem,Template:R which says:
- If the odd prime Template:Mvar does not divide any of the numerators of the Bernoulli numbers Template:Math then Template: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.Template:R
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}
- Let Template:Mvar be an odd prime and Template:Mvar an even number such that Template:Math does not divide Template:Mvar. Then for any non-negative integer Template: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 Template:Math-adic continuity.
Template:Math-adic continuityEdit
If Template:Mvar, Template:Mvar and Template:Mvar are positive integers such that Template:Mvar and Template:Mvar are not divisible by Template:Math and Template: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 Template: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 Template:Math and Template:Math, so that Template:Mvar and Template:Mvar are nonpositive and not congruent to 1 modulo Template:Math. This tells us that the Riemann zeta function, with Template:Math taken out of the Euler product formula, is continuous in the [[p-adic number|Template:Mvar-adic number]]s on odd negative integers congruent modulo Template:Math to a particular Template:Math, and so can be extended to a continuous function Template:Math for all Template:Mvar-adic integers <math>\mathbb{Z}_p,</math> the [[p-adic zeta function|Template:Mvar-adic zeta function]].
Ramanujan's congruencesEdit
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 theoremEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The von Staudt–Clausen theorem was given by Karl Georg Christian von StaudtTemplate:R and Thomas ClausenTemplate:R independently in 1840. The theorem states that for every Template:Math,
- <math> B_{2n} + \sum_{(p-1)\,\mid\,2n} \frac1p</math>
is an integer. The sum extends over all primes Template:Math for which Template:Math divides Template:Math.
A consequence of this is that the denominator of Template:Math is given by the product of all primes Template:Math for which Template:Math divides Template:Math. In particular, these denominators are square-free and divisible by 6.
Why do the odd Bernoulli numbers vanish?Edit
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 Template:Math. Doing so will show that it is an odd function for even values of Template:Math, which implies that the sum has only terms of odd index. This and the formula for the Bernoulli sum imply that Template:Math is 0 for Template:Math even and Template:Math; and that the term for Template: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 Template:Math the number Template: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 Template:Math be the number of surjective maps from Template:Math} to Template:Math}, then Template: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:Template:R
- For every Template:Math there exists a constant Template:Math (depending on Template:Math) such that Template:Math as Template:Math.
Here Template: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>
Template:Math denotes the rising factorial power in the notation of D. E. Knuth. The numbers Template:Math occur frequently in the study of the zeta function and are significant because Template:Math is a Template:Math-integer for primes Template:Math where Template:Math does not divide Template:Math. The Template:Math are called divided Bernoulli numbers.
Generalized Bernoulli numbersEdit
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 Template:Mvar-functions]] in the same way that Bernoulli numbers are related to special values of the Riemann zeta function.
Let Template:Mvar be a Dirichlet character modulo Template:Mvar. The generalized Bernoulli numbers attached to Template: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 Template:Math, we have, for any Dirichlet character Template:Mvar, that Template:Math if Template:Math.
Generalizing the relation between Bernoulli numbers and values of the Riemann zeta function at non-positive integers, one has the for all integers Template:Math:
- <math>L(1-k,\chi)=-\frac{B_{k,\chi}}k,</math>
where Template:Math is the Dirichlet Template:Mvar-function of Template:Mvar.Template:R
Eisenstein–Kronecker numberEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Eisenstein–Kronecker numbers are an analogue of the generalized Bernoulli numbers for imaginary quadratic fields.Template:R They are related to critical L-values of Hecke characters.Template:R
AppendixEdit
Assorted identitiesEdit
Template: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 Template: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 Template: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 Template:Math or Template:Math results in the Bernoulli number identity in one or another convention.
|13 = The next formula is true for Template:Math if Template:Math, but only for Template:Math if Template:Math.
- <math> \sum_{k=0}^n \binom{n}{k} \frac{B_k}{n-k+2} = \frac{B_{n+1}}{n+1} </math>
|14 = Let Template: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:Template: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 alsoEdit
- 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
NotesEdit
ReferencesEdit
BibliographyEdit
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