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In Lie theory, Bernoulli numbers appear as coefficients in the linear part of the Hausdorff series (and the recursive relation for the Dynkin Lie polynomials appearing in the Hausdorff series); this has consequences in deformation theory. The (determinant of the square root of the) inverse of its generating function appears (for variables in an adjoint representation) in an expression for the Duflo isomorphism in Lie theory and in its generalizations in knot theory etc.
In algebraic topology/cohomology Bernoulli numbers appear as the coefficients of the characteristic series of the A-hat genus (see there), and they (or equivalently, their generating functions) also appear in the expression for the Todd class.
The Bernoulli numbers are also proportional to the constant terms of the Eisenstein series and as such appear in the exponential form of the characteristic series of the Witten genus.
Finally they appear as the order of some groups in the image of the J-homomorphism (cf. Adams 65, section 2).
Of course, all of these cases are related to formal group laws. Formal groups bear also some other connections to Bernoulli numbers and generalizations like Bernoulli polynomials.
The Riemann zeta-function $\zeta$ at negative integral values is proportional to the Bernoulli numbers as
Bernoulli numbers appear also in umbral calculus. There are generalizations, for example, Bernoulli polynomials.
They also have applications in analysis (Euler-MacLaurin formula, with applications in numerical analysis).
The Bernoulli numbers $B_k$ are rational numbers given by their generating function, i.e. by the equation of functions/power series $x \mapsto f(x)$
The $k$th Bernoulli number $B_k \in \mathbb{Q}$ is, depending on convention, either equal to $\beta_k$ (or sporadically, in older literature, to $(-1)^{k-1} \beta_{2k}$). If we take generating function $x+\frac{x}{e^x-1}=\frac{x e^x}{e^x -1}=\frac{-x}{e^{-x}-1}$ this only changes the sign of $B_1$ as all other odd Bernoulli numbers (in standard convention) vanish.
The von Staudt-Clausen theorem states that
Textbook accounts;
Exposition:
Pierre Cartier, chapter 3 in: Mathemagics, Séminaire Lotharingien de Combinatoire 44 (2000), Article B44d (pdf, pdf)
John Baez, The Bernoulli numbers, 2003 expository notes (pdf)
See also:
Wikipedia, Bernoulli number
Wolfram MathWorld: Bernoulli number
Bernoulli numbers page bernoulli.org
MathOverflow Todd class and Baker-Campbell-Hausdorff, or the curious number 12
N. Durov, S. Meljanac, A. Samsarov, Z. Škoda, A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra, Journal of Algebra 309:1, pp.318-359 (2007) math.RT/0604096, MPIM2006-62
Vinay Kathotia, Kontsevich’s universal formula for deformation quantization and the Campbell-Baker-Hausdorff formula, I, math.QA/9811174
Emanuela Petracci, Functional equations and Lie algebras, PhD thesis, pdf
E. Meinrenken, Clifford algebras and Lie theory, Springer
Anton Alekseev, Bernoulli numbers, Drinfeld associators, and the Kashiwara–Vergne problem, slides, pdf
Dror Bar-Natan, Stavros Garoufalidis, Lev Rozansky, Dylan Thurston, Wheels, wheeling, and the Kontsevich integral of the unknot (q-alg/9703025)
(modified Bernoulli numbers in the universal Vassiliev invariant of the unknot)
Appearance of Bernoulli numbers in perturbative quantum field theory and string theory:
In the context of the A-hat genus the Bernoulli numbers are discussed in
Piergiulio Tempesta, Formal groups, Bernoulli-type polynomials and L-series, Comptes Rendus de l Académie des Sciences - Series I - Mathematics 07/2007; 345 doi
Stefano Marmi, Piergiulio Tempesta, Hyperfunctions, formal groups and generalized Lipschitz summation formulas, Nonlinear Analysis 03/2012; 75:1768-1777 doi
Last revised on November 21, 2020 at 23:31:08. See the history of this page for a list of all contributions to it.