# Bernoulli number

The Bernoulli numbers or Bernoulli numbers, 1, ± 12 , 16 , 0, - 130 , ... are a sequence of rational numbers that occur in mathematics in different contexts: in the expansion coefficients trigonometric , hyperbolic and other functions , in the Euler-Maclaurin formula and in number theory in connection with the Riemann zeta function . The naming of these numbers after their discoverer Jakob I Bernoulli was introduced by Abraham de Moivre .

## definition

In the mathematical specialist literature, the Bernoulli numbers are defined as three different sequences that are very closely related. There is the older notation (mainly used until the 20th century), which is denoted here with , and the two newer forms, which in this article are denoted with and and have mostly been used since the middle of the 20th century. A more precise distribution or the historical transition of the conventions is difficult to objectify, since this strongly depends on the mathematician and the area of ​​distribution of his writings. An implicit definition of Bernoulli numbers that is common nowadays is to use the coefficients of the following Taylor series either as ${\ displaystyle \ beta _ {n}}$${\ displaystyle B_ {n}}$${\ displaystyle B_ {n} ^ {\ ast}}$

${\ displaystyle {\ frac {x} {e ^ {x} -1}} = \ sum _ {k = 0} ^ {\ infty} B_ {k} {\ frac {x ^ {k}} {k! }}}$

or (by mirroring on the y-axis) as

${\ displaystyle {\ frac {x} {1-e ^ {- x}}} = \ sum _ {k = 0} ^ {\ infty} B_ {k} ^ {\ ast} {\ frac {x ^ { k}} {k!}}}$

or earlier than

${\ displaystyle {\ frac {x} {e ^ {x} -1}} = 1 - {\ frac {1} {2}} x- \ sum _ {k = 1} ^ {\ infty} (- 1 ) ^ {k} \ beta _ {k} {\ frac {x ^ {2k}} {(2k)!}}}$

to introduce. Here are the numbers and the coefficients of the series expansion or the terms of the Bernoulli number sequence. The series expansions converge for all x with Replaced by , one recognizes the validity of , i. H. the first two definitions only differ for index 1 , all others or with an odd index are zero. To distinguish clearly, the links can be referred to as those of the first type (with ) and those of the second type (with ). ${\ displaystyle B_ {k}}$${\ displaystyle B_ {k} ^ {\ ast}}$${\ displaystyle | x | <2 \ pi.}$${\ displaystyle x}$${\ displaystyle -x}$${\ displaystyle B_ {k} = (- 1) ^ {k} B_ {k} ^ {\ ast}}$${\ displaystyle B_ {k}}$${\ displaystyle B_ {k} ^ {\ ast}}$${\ displaystyle B_ {k}}$${\ displaystyle B_ {1} = - 1/2}$${\ displaystyle B_ {k} ^ {\ ast}}$${\ displaystyle B_ {1} ^ {\ ast} = 1/2}$

The older definition is based on the last row listed; in this only terms with indices occur, i. H. the terms with index 0 and 1 must be considered separately. For the remaining coefficients with an even index (precisely these are not zero) one chooses a separate definition so that they are all positive. Therefore applies${\ displaystyle k \ geq 2}$${\ displaystyle k = 2k ^ {\ prime}}$${\ displaystyle \ beta _ {k ^ {\ prime}} = (- 1) ^ {k ^ {\ prime} +1} B_ {2k ^ {\ prime}}.}$

This is exactly what Jakob I Bernoulli had done when he was first identified and thus established the older notation, although he had not yet numbered it. He discovered these numbers by considering the polynomials, which describe the sum of the powers of natural numbers from 1 to a given one with small integer exponents. E.g. ${\ displaystyle n}$

${\ displaystyle {\ begin {array} {lll} 1 ^ {\;} + 2 ^ {\;} + \ cdots + n ^ {\;} & = {\ frac {1} {2}} (n + 1) n & = {\ frac {1} {2}} n ^ {2} + {\ frac {1} {2}} n \\ 1 ^ {2} + 2 ^ {2} + \ cdots + n ^ {2} & = {\ frac {1} {6}} n (n + 1) (2n + 1) & = {\ frac {1} {3}} n ^ {3} + {\ frac {1} {2}} n ^ {2} + {\ frac {1} {6}} n \\ 1 ^ {3} + 2 ^ {3} + \ cdots + n ^ {3} & = {\ frac {1 } {4}} n ^ {2} (n + 1) ^ {2} & = {\ frac {1} {4}} n ^ {4} + {\ frac {1} {2}} n ^ { 3} + {\ frac {1} {4}} n ^ {2}, \ end {array}}}$

This ultimately leads via Faulhaber's formulas to the Euler-Maclaurin formula , in which the Bernoulli numbers play a central role. He did not prove their general values, only correctly calculated those of the smaller coefficients - his corresponding notes were published posthumously .

## Numerical values

The first Bernoulli numbers , ≠ 0 are ${\ displaystyle B_ {k}}$${\ displaystyle B_ {k} ^ {\ ast}}$

index counter denominator to 6 decimal places multiplied by  ${\ displaystyle \ mathbf {\; 2 ^ {k + 1} \! - \! 2}}$ ${\ displaystyle \ mathbf {| T_ {k-1} |}}$
0 1 1 1.000000 0
1 ± 1 2 ± 0.500000 ± 1 1
2 1 6th 0.166666 1 1
4th −1 30th −0.033333 −1 2
6th 1 42 0.023809 3 16
8th −1 30th −0.033333 −17 272
10 5 66 0.075757 155 7936
12 −691 2730 −0.253113 −2073 353792
14th 7th 6th 1.166666 38227 22368256
16 −3617 510 −7.092156 −929569 1903757312
18th 43867 798 54.971177 28820619 209865342976
20th −174611 330 −529.124242 −1109652905 29088885112832
22nd 854513 138 6192,123188 51943281731 4951498053124096
24 −236364091 2730 −86580.253113 −2905151042481 1015423886506852352
${\ displaystyle \ forall \, k \ in \ mathbb {N}: \ quad B_ {2k + 1} = 0}$
${\ displaystyle \ forall \, k \ in \ mathbb {N}: \ quad \ beta _ {2k-1} = B_ {4k-2}> 0}$
${\ displaystyle \ forall \, k \ in \ mathbb {N}: \ quad - \ beta _ {2k} = B_ {4k} <0}$

The numbers form a strictly convex (their differences grow) sequence. The denominators are always a multiple of 6, because the Clausen and von-Staudt theorem , also known as Staudt-Clausen's theorem , applies : ${\ displaystyle \ beta _ {k}}$${\ displaystyle \ beta _ {k}}$

${\ displaystyle \ forall \, k \ in \ mathbb {N} \ colon \ qquad {\ text {denominator}} (B_ {2k}) = \ prod _ {p \ in \ mathbb {P} \ atop p-1 \, | \, 2k} p}$

It is named after the independent discovery by Thomas Clausen and Karl von Staudt in 1840. The denominator is the product of all prime numbers for which it is true that the index divides. Using Fermat's Little Theorem it follows that the factor converts these rational numbers into whole numbers. ${\ displaystyle B_ {2k}}$${\ displaystyle p-1}$${\ displaystyle 2k}$${\ displaystyle 2 (2 ^ {2k} -1)}$

Even if the result of the first magnitude relatively small number of values assumes goes with increasing but more quickly to infinity than any exponential . So is z. B. ${\ displaystyle B_ {2k}}$${\ displaystyle | B_ {2k} |}$${\ displaystyle k}$

${\ displaystyle B_ {100} \ approx -2 {,} 838 \ cdot 10 ^ {78}}$ and ${\ displaystyle B_ {1000} \ approx -5 {,} 319 \ cdot 10 ^ {1769}.}$

Your asymptotic behavior can be compared with

${\ displaystyle \ beta _ {k} = | B_ {2k} | \ sim {\ frac {2 \, (2k)!} {(2 \ pi) ^ {2k}}}}$

describe, therefore the radius of convergence of the Taylor series, which were used for their definition above, is the same${\ displaystyle 2 \ pi.}$

## Recursion formulas

If one would like to describe the Bernoulli numbers of the first kind , then these Bernoulli numbers result from the recursion formula with${\ displaystyle B_ {1} = - 1/2}$${\ displaystyle B_ {k}}$${\ displaystyle n \ in \ mathbb {N}}$

${\ displaystyle B_ {n} = - {\ frac {1} {n + 1}} \ sum _ {k = 0} ^ {n-1} {n + 1 \ choose k} B_ {k}}$

and the start value . For odd indices it follows again . This formula comes from the implicit definition of the Bernoulli numbers of the first kind, which was also the most common definition until the middle of the 20th century because it has an easy-to-remember form: ${\ displaystyle B_ {0} = 1}$${\ displaystyle k \ geq 3}$${\ displaystyle B_ {k} = 0}$

${\ displaystyle \ forall \, n \ in \ mathbb {N} _ {0} \ setminus \ {1 \} \ colon \ qquad B ^ {n} = (1 + B) ^ {n},}$

which can also be written in the less common form than

${\ displaystyle \ forall \, n \ in \ mathbb {N} _ {0} \ colon \ qquad (-B) ^ {n} = (1 + B) ^ {n},}$

In these representations, powers of are to be interpreted as the correspondingly indexed Bernoulli numbers. ${\ displaystyle B}$For the Bernoulli numbers of the second kind one can use analogously

${\ displaystyle \ forall \, n \ in \ mathbb {N} _ {0} \ setminus \ {1 \} \ colon \ qquad {B ^ {\ ast}} ^ {n} = (B ^ {\ ast} -1) ^ {n}}$

as well as

${\ displaystyle \ forall \, n \ in \ mathbb {N} _ {0} \ colon \ qquad {B ^ {\ ast}} ^ {n} = (1 + B ^ {\ ast}) ^ {n} -n}$

or more elegant

${\ displaystyle \ forall \, n \ in \ mathbb {N} _ {0} \ colon \ qquad {B ^ {\ ast}} ^ {n} = (1-B ^ {\ ast}) ^ {n} }$

write and use as an inductive definition of the Bernoulli numbers of the second kind with to ${\ displaystyle n \ in \ mathbb {N}}$

${\ displaystyle B_ {n} ^ {\ ast} = {\ frac {1} {n + 1}} \ sum _ {k = 0} ^ {n-1} (- 1) ^ {n + 1-k } {n + 1 \ choose k} B_ {k} ^ {\ ast}}$

with the start value or for all as ${\ displaystyle B_ {0} ^ {\ ast} = 1}$${\ displaystyle n \ in \ mathbb {N} _ {0}}$

${\ displaystyle B_ {n} ^ {\ ast} = 1 - {\ frac {1} {n + 1}} \ sum _ {k = 0} ^ {n-1} {n + 1 \ choose k} B_ {k} ^ {\ ast}}$

## Rows with Bernoulli numbers

These numbers occur, for example, in the Taylor series of the tangent , the hyperbolic tangent or the cosecan ; in general, when a function has a closed representation where the sine function (or hyperbolic sine function) is in the denominator - i.e. H. is divided by the sum or difference of two exponential functions :

${\ displaystyle \ forall x {\ text {with}} | x | <{\ frac {\ pi} {2}} \ colon \ qquad \ tan (x) = \ sum _ {k = 1} ^ {\ infty } (- 1) ^ {k} {\ frac {2 ^ {2k} (1-2 ^ {2k})} {(2k)!}} B_ {2k} x ^ {2k-1}}$
${\ displaystyle \ forall x {\ text {with}} | x | <{\ frac {\ pi} {2}} \ colon \ qquad \ tanh (x) = \ sum _ {k = 1} ^ {\ infty } {\ frac {2 ^ {2k} (2 ^ {2k} -1)} {(2k)!}} B_ {2k} x ^ {2k-1}}$
${\ displaystyle \ forall x {\ text {with}} | x | <\ pi \ colon \ qquad \ csc (x) = \ sum _ {k = 0} ^ {\ infty} (- 1) ^ {k} {\ frac {2-2 ^ {2k}} {(2k)!}} B_ {2k} x ^ {2k-1}}$
${\ displaystyle \ forall x {\ text {with}} | x | <\ pi \ colon \ qquad \ cot (x) = \ sum _ {k = 0} ^ {\ infty} (- 1) ^ {k} {\ frac {2 ^ {2k}} {(2k)!}} B_ {2k} x ^ {2k-1}}$

Here are two non-converging asymptotic series that correspond to the trigamma function (the second derivative of the natural logarithm of the gamma function )

${\ displaystyle \ psi _ {1} (z) \ simeq \ sum _ {k = 0} ^ {\ infty} {\ frac {B_ {k} ^ {\ ast}} {z ^ {k + 1}} }, \ quad z \ to \ infty}$

and that of the natural logarithm of the gamma function

${\ displaystyle \ ln \ Gamma (x + 1) \ simeq x \ ln x-x + {\ frac {\ ln x} {2}} + \ ln {\ sqrt {2 \ pi}} + \ sum _ {k = 1} ^ {\ infty} {\ frac {B_ {2k}} {2k (2k-1) x ^ {2k-1}}}, \ quad x \ to \ infty,}$

which is known as the logarithm of the Stirling formula. This can be easily derived from the asymptotic form of the Euler-Maclaurin formula , in its symmetrical notation

${\ displaystyle \ sum _ {i = m} ^ {n} f (i) = \ sum _ {j = 0} ^ {\ infty} {\ frac {1} {j!}} \ left (B_ {j } ^ {\ ast} f ^ {(j-1)} (n) -B_ {j} f ^ {(j-1)} (m) \ right)}$

is - in which case the term that (especially for th derivation of the integral) of the function evaluated at the location means - when there is, the lower summation limit to selected and the upper summation limit with holding variable. This is one of the best known applications of Bernoulli numbers and applies to all analytic functions , even if this asymptotic expansion does not converge in most cases. ${\ displaystyle f ^ {(j-1)} (x)}$${\ displaystyle \ textstyle j \! - \! 1}$${\ displaystyle \ textstyle j \! = \! 0}$${\ displaystyle f}$${\ displaystyle x}$${\ displaystyle f (i) = \ ln i}$${\ displaystyle \ textstyle m}$${\ displaystyle \ textstyle 1}$${\ displaystyle \ textstyle n}$${\ displaystyle \ textstyle x}$${\ displaystyle f}$

## Connection with the Riemann zeta function

The following series developments provide the (in the above sense) "classic" Bernoulli numbers:

{\ displaystyle {\ begin {aligned} \ beta _ {n} = {\ frac {(2n)!} {2 ^ {2n-1} \ pi ^ {2n}}} \ zeta (2n) = {\ frac {(2n)!} {2 ^ {2n-1} \ pi ^ {2n}}} \ sum _ {k = 1} ^ {\ infty} {\ frac {1} {k ^ {2n}}} = {\ frac {2 \, (2n)!} {(2 ^ {2n} -1) \ pi ^ {2n}}} \ sum _ {k = 0} ^ {\ infty} {\ frac {1} { (2k + 1) ^ {2n}}} = {\ frac {(2n)!} {(2 ^ {2n-1} -1) \ pi ^ {2n}}} \ sum _ {k = 1} ^ {\ infty} {\ frac {(-1) ^ {k-1}} {k ^ {2n}}}. \ end {aligned}}}

The following applies to the “modern” Bernoulli numbers

{\ displaystyle {\ begin {aligned} B_ {n} ^ {\ ast} & = - {\ frac {n! \, \ cos ({\ frac {\ pi} {2}} n)} {2 ^ { n-1} \ pi ^ {n}}} \ zeta (n) = - {\ frac {n! \, \ cos ({\ frac {\ pi} {2}} n)} {2 ^ {n- 1} \ pi ^ {n}}} \ sum _ {k = 1} ^ {\ infty} {\ frac {1} {k ^ {n}}} = - {\ frac {2 \, n! \, \ cos ({\ frac {\ pi} {2}} n)} {(2 ^ {n} -1) \ pi ^ {n}}} \ sum _ {k = 0} ^ {\ infty} {\ frac {1} {(2k + 1) ^ {n}}} \\ & = {\ frac {n! \, \ cos ({\ frac {\ pi} {2}} n)} {(2 ^ { n-1} -1) \ pi ^ {n}}} \ sum _ {k = 1} ^ {\ infty} {\ frac {(-1) ^ {k}} {k ^ {n}}}, \ end {aligned}}}

wherein in the case of the more recent definition for n = 1 undefined terms of shape occur, but according to the rule of de l'Hospital because of the first-order pole of the Riemann zeta function at 1 (or in the last representation the term cancel in the denominator), and thus correctly deliver the value . ${\ displaystyle {\ tfrac {0} {0}}}$${\ displaystyle \ textstyle \ lim _ {n \ to 1} \ cos ({\ frac {\ pi} {2}} n) = \ lim _ {n \ to 1} {\ frac {\ pi} {2} } (1-n)}$${\ displaystyle \ textstyle 2 ^ {n-1} -1}$${\ displaystyle {\ tfrac {1} {2}}}$

For the Bernoulli numbers of the second kind there is also the concise representation

${\ displaystyle B_ {n} ^ {\ ast} = - n \, \ zeta (1-n) \ quad \ forall \, n \ in \ mathbb {N} _ {0},}$

so that the entire theory of the Riemann zeta function for characterizing the Bernoulli numbers is available.

For example, the product representation of the Riemann zeta function and the above series expansion of the Bernoulli numbers result in the following representation:

${\ displaystyle \ beta _ {n} = {\ frac {2 \, (2n)!} {(2 \ pi) ^ {2n}}} \ \ prod _ {p \ in \ mathbb {P}} \ left (1 - {\ frac {1} {p ^ {2n}}} \ right) ^ {- 1} = {\ frac {2 \, (2n)!} {(2 \ pi) ^ {2n}}} \ {\ frac {1} {\ left (1 - {\ frac {1} {2 ^ {2n}}} \ right) \ left (1 - {\ frac {1} {3 ^ {2n}}} \ right) \ left (1 - {\ frac {1} {5 ^ {2n}}} \ right) \ cdots}}}$ .

Here the product extends over all prime numbers (see also Euler product of the Riemann zeta function).

## Integral representations

There are many improper integrals with sums or differences of two exponential functions in the denominator of the integrand whose values ​​are given by Bernoulli numbers. Some simple examples are

${\ displaystyle \ forall \, n \ in \ mathbb {N} \; \ forall \, a \ in \ mathbb {R} ^ {+} \ colon \ qquad \ int _ {0} ^ {\ infty} {\ frac {x ^ {2n-1}} {e ^ {ax} -e ^ {- ax}}} \; {\ text {d}} x = {\ frac {2 ^ {2n} -1} {4 }} \ beta _ {n} \ left ({\ frac {\ pi} {a}} \ right) ^ {2n}}$
${\ displaystyle \ forall \, n \ in \ mathbb {N} \; \ forall \, a \ in \ mathbb {R} ^ {+} \ colon \ qquad \ int _ {0} ^ {\ infty} {\ frac {x ^ {2n-1}} {e ^ {ax} -1}} \; {\ text {d}} x = {\ frac {\ beta _ {n}} {4}} \ left ({ \ frac {2 \ pi} {a}} \ right) ^ {2n}}$
${\ displaystyle \ forall \, n \ in \ mathbb {N} \; \ forall \, a \ in \ mathbb {R} ^ {+} \ colon \ qquad \ int _ {0} ^ {\ infty} {\ frac {x ^ {2n-1}} {e ^ {ax} +1}} \; {\ text {d}} x = {\ frac {2 ^ {2n} -1} {2n}} \ beta _ {n} \ left ({\ frac {\ pi} {a}} \ right) ^ {2n}}$

but also

${\ displaystyle \ forall \, n \ in \ mathbb {N} \; \ forall \, a \ in \ mathbb {R} ^ {+} \ colon \ qquad \ int _ {0} ^ {1} (\ ln x) ^ {2n-2} \ ln (1-x ^ {a}) {\ frac {1} {x}} \; {\ text {d}} x = {\ frac {- (2 \ pi) ^ {2n-1} \ beta _ {n}} {4n (2n-1) a ^ {2n-1}}}}$

out.

## Bernoulli polynomials

The graphs of the Bernoulli polynomials of degrees 1 to 6

The Bernoulli polynomial is a mapping for each and is fully characterized by the following recursion equations: For we set ${\ displaystyle n \ in \ mathbb {N} _ {0}}$ ${\ displaystyle {\ text {B}} _ {n} \ colon [0,1] \ rightarrow \ mathbb {R}}$${\ displaystyle n = 0}$

${\ displaystyle {\ text {B}} _ {0} (x): = 1}$

and for the -th Bernoulli polynomial results uniquely from the two conditions ${\ displaystyle n \ geq 1}$${\ displaystyle n}$${\ displaystyle {\ text {B}} _ {n}}$

${\ displaystyle {\ text {B}} _ {n} (x) = n \ int {\ text {B}} _ {n-1} (x) \, {\ text {d}} x}$

and

${\ displaystyle \ int _ {0} ^ {1} {\ text {B}} _ {n} (x) \, {\ text {d}} x = 0}$

recursively from the previous one. As the sum of the powers of written, the expression for the -th polynomial is ${\ displaystyle x}$${\ displaystyle n}$

${\ displaystyle {\ text {B}} _ {n} (x) = \ sum \ limits _ {k = 0} ^ {n} {n \ choose k} B_ {k} \, x ^ {nk}, }$

where here again denote the Bernoulli numbers of the first kind. This form follows directly from the symbolic formula ${\ displaystyle B_ {k}}$

${\ displaystyle {\ text {B}} _ {n} (x) = (B + x) ^ {n}}$

in which the powers of are interpreted as the corresponding nth Bernoulli number . The first Bernoulli polynomials are ${\ displaystyle B}$${\ displaystyle B_ {n}}$

${\ displaystyle {\ text {B}} _ {0} (x) = 1}$
${\ displaystyle {\ text {B}} _ {1} (x) = x - {\ tfrac {1} {2}}}$
${\ displaystyle {\ text {B}} _ {2} (x) = x ^ {2} -x + {\ tfrac {1} {6}}}$
${\ displaystyle {\ text {B}} _ {3} (x) = x ^ {3} - {\ tfrac {3} {2}} x ^ {2} + {\ tfrac {1} {2}} x}$
${\ displaystyle {\ text {B}} _ {4} (x) = x ^ {4} -2x ^ {3} + x ^ {2} - {\ tfrac {1} {30}}}$
${\ displaystyle {\ text {B}} _ {5} (x) = x ^ {5} - {\ tfrac {5} {2}} x ^ {4} + {\ tfrac {5} {3}} x ^ {3} - {\ tfrac {1} {6}} x}$
${\ displaystyle {\ text {B}} _ {6} (x) = x ^ {6} -3x ^ {5} + {\ tfrac {5} {2}} x ^ {4} - {\ tfrac { 1} {2}} x ^ {2} + {\ tfrac {1} {42}}}$

These polynomials are symmetrical around , more precisely ${\ displaystyle {\ tfrac {1} {2}}}$

${\ displaystyle {\ text {B}} _ {k} ({\ tfrac {1} {2}} + x) = (- 1) ^ {k} {\ text {B}} _ {k} ({ \ tfrac {1} {2}} - x).}$

Their constant terms are the Bernoulli numbers of the first kind, that is

${\ displaystyle {\ text {B}} _ {k} (0) = B_ {k},}$

the Bernoulli numbers of the second kind are obtained from

${\ displaystyle {\ text {B}} _ {k} (1) = B_ {k} ^ {\ ast}}$

and finally applies

${\ displaystyle {\ text {B}} _ {k} ({\ tfrac {1} {2}}) = - (1-2 ^ {1-k}) B_ {k} ^ {\ ast} = - (1-2 ^ {1-k}) B_ {k}}$

in the middle of the interval. The kth Bernoulli polynomial has fewer than k zeros in the whole interval for k> 5 and two for even n ≠ 0 and the three zeros for odd n ≠ 1 in the interval . Let be the set of zeros of these polynomials. Then ${\ displaystyle \ mathbb {R}}$${\ displaystyle 0, {\ tfrac {1} {2}}, 1}$${\ displaystyle [0,1]}$${\ displaystyle R (n) = \ {x \ in \ mathbb {R} \ colon {\ text {B}} _ {n} (x) = 0 \}}$

${\ displaystyle - {\ tfrac {1} {4}} | R (n) | + {\ tfrac {3} {4}} \ leq \ min R (n) \ leq \ max R (n) \ leq { \ tfrac {1} {4}} | R (n) | + {\ tfrac {1} {4}}}$

for all n ≠ 5 and n ≠ 2 and it holds

${\ displaystyle \ lim _ {n \ to \ infty} {\ frac {| R (n) |} {n}} = {\ frac {2} {\ pi e}} \ approx 0 {,} 2342,}$

where the function applied to a set indicates the number of elements. ${\ displaystyle | \ cdot |}$

The function values ​​of the Bernoulli polynomials in the interval [0,1] are through for an even index

${\ displaystyle - | B_ {k} | \ leq {\ text {B}} _ {k} (x) \ leq | B_ {k} |}$

and for odd index (but not sharp) through ${\ displaystyle \ not = 1}$

${\ displaystyle - {\ frac {2 \ zeta (k) k!} {(2 \ pi) ^ {k}}} <{\ text {B}} _ {k} (x) <{\ frac {2 \ zeta (k) k!} {(2 \ pi) ^ {k}}}}$

limited.

They also satisfy the equation

${\ displaystyle {\ text {B}} _ {k} (x + 1) = {\ text {B}} _ {k} (x) + kx ^ {k-1}}$

if they are continued analytically and the sum of the power of the first n natural numbers can be expressed as ${\ displaystyle \ mathbb {R}}$

${\ displaystyle \ sum _ {j = 1} ^ {n} j ^ {k} = \ int _ {0} ^ {n + 1} B_ {k} (t) \, {\ text {d}} t = {\ frac {{\ text {B}} _ {k + 1} (n + 1) - {\ text {B}} _ {k + 1} (0)} {k + 1}}}$

describe. The index shift from to on the right-hand side of the equation is necessary here because historically the Bernoulli poynomials were "incorrectly" attached to the Bernoulli numbers of the first type (and not the second type) and thus obtained instead of the summands in the above Bernoulli poynomials, what exactly results in the value too little here (the last term of the sum on the left side) and therefore this index has to run “one further” on the right side. ${\ displaystyle n}$${\ displaystyle n + 1}$${\ displaystyle {\ tfrac {k} {2}} {n ^ {k}}}$${\ displaystyle - {\ tfrac {k} {2}} n ^ {k}}$${\ displaystyle n ^ {k}}$

## Bernoulli numbers in algebraic number theory

Staudt's theorem:

${\ displaystyle \ forall \, p \ in \ mathbb {P} \; \ forall \, n \ in \ mathbb {N} {\ text {with}} (p-1) \, | \, 2n \; \ colon \ qquad pB_ {2n} \ equiv -1 {\ pmod {p}}}$

As a sentence from Staudt-Clausen is also the statement

${\ displaystyle B_ {2n} + \! \ sum _ {p \ in \ mathbb {P} \ atop p-1 \, | \, 2n} \! {\ frac {1} {p}} \ quad \ in \; \ mathbb {Z}}$

known, which is somewhat stronger than the previous theorem by Clausen and von Staudt for characterizing the denominator. The sequence of the whole numbers determined in this way for an even index is . ${\ displaystyle 1,1,1,1,1,1,2, -6.56, -528.6193, \ ldots}$

Kummer's congruence:

${\ displaystyle \ forall \, p \ in \ mathbb {P} \; \ forall \, n \ in \ mathbb {N} {\ text {with}} (p-1) \ not | \, 2n \; \ colon \ qquad {\ frac {B_ {2n + p-1}} {2n + p-1}} \ equiv {\ frac {B_ {2n}} {2n}} {\ pmod {p}}}$

An odd number is called regular prime , if they do not counter the Bernoulli numbers with shares. Kummer showed that this condition is equivalent to not dividing the class number of the p-th pitch field . In 1850 he was able to prove that Fermat's great theorem , namely has for no solutions in , holds for all exponents that are a regular prime number. For example, by checking the Bernoulli numbers up to index 94, Fermat's great theorem was proven for all other exponents with the exception of the exponents 37, 59, 67 and 74 . ${\ displaystyle p \ in \ mathbb {P}}$${\ displaystyle B_ {2n}}$${\ displaystyle 2n \ leq p-3}$${\ displaystyle p}$ ${\ displaystyle h _ {\ mathbb {Q} (\ zeta _ {p})}}$ ${\ displaystyle \ mathbb {Q} (\ zeta _ {p})}$${\ displaystyle a ^ {p} + b ^ {p} = c ^ {p}}$${\ displaystyle p> 2}$${\ displaystyle \ mathbb {N}}$${\ displaystyle p}$${\ displaystyle \ leq 100}$

## Tangent numbers and applications in combinatorics

If one considers the Euler's numbers and the Taylor expansion of the tangent function, then one can define the tangent numbers implicitly

${\ displaystyle \ forall x {\ text {with}} | x | <{\ frac {\ pi} {2}} \ colon \ qquad \ tan (x) = \ sum _ {k = 1} ^ {\ infty } (- 1) ^ {k} {\ frac {2 ^ {2k} (1-2 ^ {2k})} {(2k)!}} B_ {2k} x ^ {2k-1} = \ sum _ {k = 1} ^ {\ infty} (- 1) ^ {k} {\ frac {T_ {2k-1}} {(2k-1)!}} x ^ {2k-1}}$

and still set for index zero . So you have the transformation ${\ displaystyle T_ {0} = 1}$

${\ displaystyle \ forall \, n \ in \ mathbb {N} \ colon \ quad T_ {n-1} = - {\ frac {2 ^ {n} (2 ^ {n} -1)} {n}} B_ {n}}$

which generates this sequence of integers from the Bernoulli numbers of the first kind:

${\ displaystyle (T_ {n}) _ {n \ in \ mathbb {N} _ {0}} = (1, -1,0,2,0, -16,0,272,0, -7936, \ ldots) }$

Since the choice of sign in the implicit definition is completely arbitrary, one can just as justifiably use

${\ displaystyle \ forall \, n \ in \ mathbb {N} \ colon \ quad T_ {n-1} ^ {\ ast} = \ mp {\ frac {2 ^ {n} (2 ^ {n} -1 )} {n}} B_ {n} ^ {\ ast}}$

define the tangent numbers, with the consequence

${\ displaystyle (T_ {n} ^ {\ ast}) _ {n \ in \ mathbb {N} _ {0}} = (\ mp 1, \ mp 1.0, \ pm 2.0, \ mp 16 , 0, \ pm 272.0, \ mp 7936, \ ldots)}$

and has indexes for all ${\ displaystyle T_ {n} ^ {\ ast} = \ pm 2 ^ {n + 1} (2 ^ {n + 1} -1) \ zeta (-n).}$

In any case, with the exception of, all numbers with an even index are zero and those with an odd index have alternating signs. ${\ displaystyle T_ {0}}$

The values are now exactly the number of alternating permutations of an elementary set. Further information on the direct determination of tangent numbers can be found in the article Euler's numbers . ${\ displaystyle 2 | T_ {2k + 1} |}$${\ displaystyle 2k + 1}$

In combinatorics, the Bernoulli numbers of the second kind can also be represented by the Stirling numbers of the second kind as ${\ displaystyle \ textstyle \ left \ {{n \ atop k} \ right \}}$

${\ displaystyle \ forall \, n \ in \ mathbb {N} _ {0} \ colon \ quad B_ {n} ^ {\ ast} = \ sum _ {k = 0} ^ {n} (- 1) ^ {k} {\ frac {k!} {k + 1}} \ left \ {{n \ atop k} \ right \}}$

The values are also known as Worpitzky numbers . Another connection arises from the generating power series of the Stirling polynomials with paths ${\ displaystyle k! \ left \ {{n \ atop k} \ right \}}$ ${\ displaystyle S_ {k} (x)}$${\ displaystyle k \ in \ mathbb {N} _ {0}}$

${\ displaystyle \ sum _ {k = 0} ^ {\ infty} {\ frac {S_ {k} (x)} {k!}} t ^ {k} = \ left ({\ frac {t} {1 -e ^ {- t}}} \ right) ^ {x + 1}}$

with the Stirling numbers of the first kind to ${\ displaystyle \ textstyle \ left [{n \ atop \ ell} \ right]}$

${\ displaystyle S_ {k} (m) = {\ frac {(-1) ^ {k}} {m \ choose k}} \ left [{m + 1 \ atop m \! + \! 1 \! - \! k} \ right] \ qquad {\ text {for}} m \ in \ mathbb {N} _ {0}, \; k \ leq m + 1,}$

which one could define as negative . Hence the Bernoulli numbers of the second kind are also the values ​​of the sterling polynomials at zero ${\ displaystyle \ ell}$

${\ displaystyle S_ {k} (0) = B_ {k} ^ {\ ast}}$

due to the same formal power series.

## Algebraic topology

Here in the article, the Bernoulli numbers were initially defined arbitrarily by means of generating power series. The formal power series of also occurs directly when determining the Todd class of a vector bundle in a topological space : ${\ displaystyle {\ tfrac {x} {1-e ^ {- x}}}}$ ${\ displaystyle E}$ ${\ displaystyle X}$

${\ displaystyle \ operatorname {td} (E) = \ prod _ {i \ in \ mathbb {N}} {\ frac {c_ {i}} {1-e ^ {- c_ {i}}}} = \ prod _ {i \ in \ mathbb {N}} \; \ sum _ {k = 0} ^ {\ infty} B_ {k} ^ {\ ast} {\ frac {c_ {i} ^ {k}} { k!}}}$

where are the cohomology classes of . If is finite dimensional, then is a polynomial. The Bernoulli numbers of the second kind naturally "count" certain topological objects. This formal power series is also reflected in the L gender or Todd gender of the characteristic power series of an orientable manifold . ${\ displaystyle c_ {i}}$${\ displaystyle E}$${\ displaystyle X}$${\ displaystyle td (E)}$

## literature

• Jakob Bernoulli : Ars conjectandi , opus posthumum. (Art of guessing, work left behind), Basileæ (Basel) 1713 (Latin).
• Julius Worpitzky : Studies on Bernoulli and Euler's numbers . Crelles Journal 94, 1883, pp. 203-232.
• Senon I. Borewicz , Igor R. Šafarevič : Number theory. Birkhäuser Verlag Basel, 1966, chap. 5, § 8, pp. 408-414.
• Jürgen Neukirch : Algebraic number theory. Springer-Verlag, 1992.
• Kenneth F. Ireland, Michael Rosen : A Classical Introduction to Modern Number Theory. Graduate Texts in Mathematics, Vol. 84, Springer-Verlag, 2nd edition 1990, chap. 15, pp. 228-248.
• IS Gradshteyn, IM Ryzhik : Table of Integrals, Series and Products. Academic Press, 4th edition 1980, ISBN 0-12-294760-6 , chap. 9.6.
• Ulrich Warnecke: For the polynomial representation of for any${\ displaystyle \ sum _ {\ nu = 1} ^ {n} \ nu ^ {k}}$${\ displaystyle k \ in \ mathbb {N}.}$ In: Mathematical semester reports. Volume XXX / 1983, pp. 106-114.

## swell

1. ^ JC Kluyver: The Staudt-Clausen'sche theorem. Math. Ann. Vol. 53, (1900), pp. 591-592.
2. ^ W. Gröbner and N. Hofreiter: Integraltafel . Second part: definite integrals. 5. verb. Edition, Springer-Verlag, 1973.
3. ^ John H. Conway , Richard K. Guy : The Book of Numbers. Springer-Verlag, 1996, ISBN 0-387-97993-X , chap. 4, pp. 107-109.
4. JM Borwein , PB Borwein , K. Dilcher: Pi, Euler Numbers, and Asymptotic Expansions. AMM, Vol. 96, No. 8, (Oct. 1989), p. 682.
5. ^ Henry Wadsworth Gould: Combinatorial identities. Morgantown, W Va, 1972.
6. K. Reillag, J. Gallier: Complex Algebraic Geometry. CIS 610, Lecture Notes, Fall 2003 - Spring 2004, Chap 3, pp. 209–220 (online).