Bernoulli Numbers

 

Title: Bernoulli Numbers
Authors: Lukas Bulwahn (lukas /dot/ bulwahn /at/ gmail /dot/ com) and Manuel Eberl
Submission date: 2017-01-24
Abstract:

Bernoulli numbers were first discovered in the closed-form expansion of the sum 1m + 2m + … + nm for a fixed m and appear in many other places. This entry provides three different definitions for them: a recursive one, an explicit one, and one through their exponential generating function.

In addition, we prove some basic facts, e.g. their relation to sums of powers of integers and that all odd Bernoulli numbers except the first are zero. We also prove the correctness of the Akiyama–Tanigawa algorithm for computing Bernoulli numbers with reasonable efficiency, and we define the periodic Bernoulli polynomials (which appear e.g. in the Euler–MacLaurin summation formula and the expansion of the log-Gamma function) and prove their basic properties.

BibTeX:
@article{Bernoulli-AFP,
  author  = {Lukas Bulwahn and Manuel Eberl},
  title   = {Bernoulli Numbers},
  journal = {Archive of Formal Proofs},
  month   = jan,
  year    = 2017,
  note    = {\url{http://isa-afp.org/entries/Bernoulli.shtml},
            Formal proof development},
  ISSN    = {2150-914x},
}
License: BSD License
Used by: Stirling_Formula
Status: [ok] This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.