Bernoulli number

Bernoulli number
Name	Bernoulli number
First defined	17th century
Field	Number theory
Notable for	Series expansions, zeta values, Euler–Maclaurin formula

Bernoulli number Bernoulli numbers are a sequence of rational numbers that arise in the study of power series, finite sums, and special values of zeta and L-functions. They appear in the Euler–Maclaurin summation formula, Taylor expansions of trigonometric and hyperbolic functions, and formulae for sums of powers of consecutive integers. Bernoulli numbers connect works of Jakob Bernoulli, Leonhard Euler, and later developments by Adrien-Marie Legendre and Bernhard Riemann.

Definition

Bernoulli numbers are defined implicitly by the exponential generating function relating to the function exp(x) and the formal power series used by Jakob Bernoulli and studied by Leonhard Euler, Joseph-Louis Lagrange, Augustin-Louis Cauchy, and Niels Henrik Abel. The classical definition uses the relation between the generating function and the formal series for x/(e^x − 1), a form that links to expansions computed by John Wallis, James Stirling, and later systematized by Adrien-Marie Legendre and Carl Friedrich Gauss.

Properties and identities

Bernoulli numbers satisfy many algebraic and arithmetic identities explored by Leonhard Euler, Gottfried Wilhelm Leibniz, Émile Borel, and Srinivasa Ramanujan. They have symmetry properties in even and odd indices discovered by Jakob Bernoulli and clarified by Adrien-Marie Legendre and Bernhard Riemann. Relations to the Riemann zeta function at integer arguments were established by Leonhard Euler and later used by Bernhard Riemann in analytic continuation. Congruences for Bernoulli numbers involve results of Ernst Kummer, Richard Dedekind, and Helmut Hasse, and link to cyclotomic fields studied by Kummer and Ernst Eduard Kummer.

Generating functions

The exponential generating function historically analyzed by Jakob Bernoulli, Leonhard Euler, and Adrien-Marie Legendre expresses Bernoulli numbers in terms of e^x and rational functions, with alternative formulations studied by Joseph Fourier, Lord Kelvin, and Peter Gustav Lejeune Dirichlet. Generating functions connect to Fourier series expansions used by Joseph Fourier and to modular forms investigated by Srinivasa Ramanujan and Bernhard Riemann.

Computation and algorithms

Algorithms for computing Bernoulli numbers have been advanced by computational mathematicians connected to Ada Lovelace-era computation ideas, later refined by Edmond Halley, Charles Babbage, D. H. Lehmer, Donald Knuth, and modern implementations leveraging algorithms from Peter Borwein, Richard Brent, and researchers at institutions like Massachusetts Institute of Technology and University of Cambridge. Fast multiplication and modular techniques draw on work by Andrey Kolmogorov-era numerical analysts and contemporary complexity theory contributors such as Leslie Lamport and Stephen Cook.

Applications

Bernoulli numbers appear in formulae for sums of powers, asymptotic expansions in analysis credited to Brook Taylor and Colin Maclaurin, and in special value evaluations related to Leonhard Euler and Bernhard Riemann. They are used in numerical quadrature methods attributable to the development of the Euler–Maclaurin formula by Leonhard Euler and Colin Maclaurin, and in perturbative expansions influential in mathematical physics communities associated with Paul Dirac, Richard Feynman, and later work in quantum field theory.

History and attribution

Historical development involves Jakob Bernoulli who first presented sums of powers, with systematic study by Leonhard Euler and subsequent refinements by Adrien-Marie Legendre, Niels Henrik Abel, and Bernhard Riemann. Later contributions and congruence relations were elaborated by Ernst Kummer, Richard Dedekind, and modern expositors at institutions such as École Polytechnique and University of Göttingen.

Category:Number theory