Identity #0102 with Bernoulli numbers
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HTML paper from
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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