Identities with Bernoulli numbers
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Personal Journal of
Yuri MATIYASEVICH.
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Last modified on August 30, 2006.
Notation used
Bernoulli
numbers are famous for the numerous identities in which they occur.
I have
found some such identities which I believe to be new but I cannot tell
it for sure.
If you recognize that an identitiy below has been already
published elsewhere, please, inform me.
Most of the identities below have been found with the aid of
MATHEMATICA.
Summation indices j, k, l range over positive (even) integers.
Identity #0102
For even m>2
SUM_{k+l=m} (B_{k}/k)B_{l} 

SUM_{k+l=m} C(m,k)(B_{k}/k)B_{l} 

B_{m}H_{m}=0

This identity has related identites, known before, and generalization.
The identity was put on WWW on Feb. 28, 1997.
Identity #0202
For even m>2
(m+2)SUM_{k+l=m} B_{k}B_{l} 

2 SUM_{k+l=m} C(m+2,k)B_{k}B_{l} 

m(m+1)B_{m}=0

The identity was put on WWW on Feb. 28, 1997.
This identity was generalized by Hao Pan and Zhiwei Sun in
arXiv:math.NT/0407363 to Bernoulli polynomials.