Identity #0102 with Bernoulli numbers
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Personal Journal of
Yuri MATIYASEVICH.
Last modification done on August 30, 2006.
Notation used
This identity can be viewed as an "interpolation"
between two identities
known before. Namely, for even m>2 we have:
Euler identity:
SUM_{k+l=m}
C(m,k)B_{k}B_{l} 

(m+1)B_{m}=0

Identity #0102
(found by Yuri Matiyasevich):
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

Miki's identity:
SUM_{k+l=m}
(B_{k}/k)(B_{l}/l) 

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

2(B_{m}/m)H_{m}=0

The latter identity was found by
Hiroo Miki in
A relation between Bernoulli numbers,
J. of Number Theory, vol. 10 (1978), no. 3, pp. 297302;
MR: 80a:10024.
Another proof and generalizations of Miki's identity were given in the following works:

Katsumi Shiratani and Sayoko Yokoyama in
An application of padic convolutions,
Memoirs of the Faculty of Science. Kyushu University. Series A. Mathematics,
vol. 36 (1982), no. 1, pp. 7383;
MR: 83m:10011.
 I. M. Gessel, On Miki's identities for Bernoulli numbers, J. Number Theory, vol. 110} (2005), 7582,
http://people.brandeis.edu/~gessel/homepage/papers/miki.pdf
 G. V. Dunne and C. Schubert, Bernoulli number identities from quantum field theory,
arXiv:math.NT/0406610.
Identity #0102 was later proved and generalized in the following works:

H. Pan and ZW. Sun, New identities involving Bernoulli and Euler polynomials, J. Comb. Theory, Ser. A
vol. 113(1) (2006), 156175.
 H. Gopalkrishna Gadiyar and R. Padma, A Comment on Matiyasevich's Identity #0102 with Bernoulli Numbers,
arXiv:math.NT/0608675
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