Bernoulli Number- Bernoulli Number - What is it?

Bernoulli Number

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From Wikipedia, the free encyclopedia

In mathematics, the Bernoulli numbers are a sequence of rational numbers with deep
connections to number theory. They are closely related to the values of the Riemann zeta function at negative integers.

In Europe, they were first studied by Jakob
Bernoulli, after whom they were named by Abraham de Moivre. In
Japan, perhaps earlier, they were independently discovered by Seki Kōwa. They appear in the Taylor series expansions of
the tangent
and hyperbolic
tangent functions, in the Euler–Maclaurin formula,
and in expressions for certain values of the Riemann zeta function.

In note G
of Ada Lovelace's notes on the Analytical
Engine from 1842, Lovelace describes an algorithm
for generating Bernoulli numbers with Babbage's machine ^{[~ 1]}.
As a result, the Bernoulli numbers have the distinction of being the
subject of the first computer program.

Contents

• 1 Introduction
  
  
  
  • 1.1 Generating functions
  
  
  • 1.2 Other forms and conventions
    
  
  
• 2 Values of the Bernoulli numbers


• 3 Efficient computation of Bernoulli numbers


• 4 Reconstruction of 'Summae Potestatum'


• 5 Different viewpoints and conventions


• 6 Application of the Bernoulli number


• 7 Combinatorial definitions
  
  
  
  • 7.1 Connection with the Worpitzky number
  
  
  • 7.2 Connection with the Stirling set number
  
  
  • 7.3 Connection with the Stirling cycle number
  
  
  • 7.4 Connection with the Eulerian number
    


• 8 Asymptotic approximation


• 9 Integral representation and continuation


• 10 The relation to the Euler numbers and π


• 11 An algorithmic view: the Seidel triangle


• 12 A combinatorial view: alternating permutations


• 13 Generalizations by polynomials


• 14 Arithmetical properties of the Bernoulli numbers
  
  
  
  • 14.1 Von Staudt–Clausen theorem
  
  
  • 14.2 Why do the odd Bernoulli numbers vanish?
  
  
  • 14.3 p-adic continuity
  
  
  • 14.4 Bernoulli numbers and the Riemann hypothesis
  
  
  • 14.5 Ramanujan's congruences
    


• 15 Use of Bernoulli numbers in topology


• 16 Assorted identities


• 17 See also


• 18 References


• 19 External links