> Hello, I would like to know when the two different
> definitions of the Bernoulli's numbers were established,
> and if there is a special reason
> for the 'persistance' of these two definitions.
I know three: the b_n, the B_2n, and the two signs in
B_1 = \pm 1/2. The reason for throwing out the B. #s
with odd index is clear: why would anyone want to
denote 0 by B_3, B_5, B_7 ...? The reason for not
throwing them out is more convincing, however:
B. #s are a special case of generalized Bernoulli numbers
introduced by Leopoldt, and there the vanishing of
certain numbers depends on the parity of the character.
There's a similar problem with counting the weight
of modular forms, BTW.
> Are these numbers used in some physical theories ?
Probably, given that modular forms start appearing there.
> Are there a lot of
> theorems using these numbers (except of course the definitions) ?
You might want to check out the following:
Dilcher, Karl (ed.); Skula, Ladislav (ed.); Slavutskij, Ilya Sh.(ed.)
Bernoulli numbers. Bibliography (1713-1990). Enlarged ed.
Queens Papers in Pure and Applied Mathematics. 87. Kingston (Canada):
Queen's University, iv, 175 p. (1991).
This is an enlarged and corrected version of the
bibliography published in 1987 (see Zbl. 637.10001).
It contains 1.956 citations to publications connected
with Bernoulli numbers and related number-theoretic oriented
topics, alphabetically ordered with respect to authors and
with a useful index.
best,
franz