the odd Bernoulli numbers are zero
Recall that, for , the Bernoulli numbers are defined as the coefficients in the Taylor expansion:
(1) |
Just to name a few:
Lemma.
If is odd then .
Proof.
From the right hand side of (1) we extract the term corresponding to :
(2) |
Thus:
(3) |
and the left hand side can be rewritten as:
(4) |
Hence, if one replaces by then (4) is unchanged. Since (4) is the left hand side of (3), the quantity
is also unchanged when is exchanged by , and so we must have for . We conclude that if and is odd, . ∎
Title | the odd Bernoulli numbers are zero |
---|---|
Canonical name | TheOddBernoulliNumbersAreZero |
Date of creation | 2013-03-22 15:12:04 |
Last modified on | 2013-03-22 15:12:04 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 5 |
Author | alozano (2414) |
Entry type | Theorem |
Classification | msc 11B68 |
Related topic | KummersCongruence |
Related topic | CongruenceOfClausenAndVonStaudt |