> 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.
Actually there are more than two notations flourishing. I've seen 1)-3)
below, and think 4) probably exists (ones not mentioned are 0):
1) B0 = 1, B1 = 1/2, B2 = 1/6, B4 = -1/30, B6 = 1/42, B8 = -1/30 ...
2) B0 = 1, B1 =-1/2, B2 = 1/6, B4 = -1/30, B6 = 1/42, B8 = -1/30
3) B0 = 1, B1 = 1/6, B2 = 1/30, B3 = 1/42, B4 = +1/30
4) B0 = 1, B1 = 1/6, B2 = -1/30, B3 = 1/42, B4 = -1/30 ...
In my view, 1) is the best - it differs only in the sign of B1
from 2), the one Abramwotz and Stegun use, which seems to be becoming
the standard. The older notation 3) takes absolute values and halves
the indices merely for the sake of avoiding negative and zero numbers -
unfortunately it still has some currency.
Since one can always display the first few terms of a series
explicitly, the difference between 1) and 2) doesn't really matter;
but the difference between these and the other two is important -
the formulae are all much simpler and more natural if you use 1) or 2).
> Are these numbers used in some physical theories ? Are there a lot of
> theorems using these numbers (except of course the definitions) ?
The big theorem is Faulhaber's summation formula - if f(x) is a
polynomial whose integral is F(x), then
f(1) + f(2) + ... + f(n) = "F(n+B) - F(B)"
where an expression like "F(B)" means
a + bB1 + cB2 + dB3 + ...
if
F(t) = a + bt + c.t^2 + d.t^3 + ... ,
and I'm using the numbering 1). [If you used 2), it would be
"F(n-B) - F(-B)", which is only slightly worse.]
Try this yourself, using the familiar formulae for
1 + 2 + 3 + ... + n and 1^2 + 2^2 + 3^2 + ... + n^2.
There's a proof of Faulhaber's formula in "The Book of Numbers".
John Conway