Identity #0102 with Bernoulli numbers

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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_kB_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_mH_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. 297-302; 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 p-adic convolutions, Memoirs of the Faculty of Science. Kyushu University. Series A. Mathematics, vol. 36 (1982), no. 1, pp. 73-83; MR: 83m:10011.
I. M. Gessel, On Miki's identities for Bernoulli numbers, J. Number Theory, vol. 110} (2005), 75-82, 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 Z-W. Sun, New identities involving Bernoulli and Euler polynomials, J. Comb. Theory, Ser. A vol. 113(1) (2006), 156-175.
H. Gopalkrishna Gadiyar and R. Padma, A Comment on Matiyasevich's Identity #0102 with Bernoulli Numbers, arXiv:math.NT/0608675

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