Identity #0102 with Bernoulli numbers
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Last modification done on August 30, 2006.
This identity can be viewed as an "interpolation"
between two identities
known before. Namely, for even m>2 we have:
(found by Yuri Matiyasevich):
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;
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;
- I. M. Gessel, On Miki's identities for Bernoulli numbers, J. Number Theory, vol. 110} (2005), 75-82,
- G. V. Dunne and C. Schubert, Bernoulli number identities from quantum field theory,
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,
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