Definitions
Many characterizations of the Bernoulli numbers have been found in the last 300 years, and each could be used to introduce these numbers. Here only four of the most useful ones are mentioned:
- a recursive equation,
- an explicit formula,
- a generating function,
- an algorithmic description.
For the proof of the equivalence of the four approaches the reader is referred to mathematical expositions like (Ireland & Rosen 1990) or (Conway & Guy 1996).
Unfortunately in the literature the definition is given in two variants: Despite the fact that Bernoulli defined B1 = 1/2 (now known as "second Bernoulli numbers"), some authors set B1 = −1/2 ("first Bernoulli numbers"). In order to prevent potential confusions both variants will be described here, side by side. Because these two definitions can be transformed simply by into the other, some formulae have this alternatingly (-1)n-term and others not depending on the context, but it is not possible to decide in favor of one of these definitions to be the correct or appropriate or natural one (for the abstract Bernoulli numbers).
Read more about this topic: Bernoulli Number
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