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On Miki’s identity for Bernoulli numbers. (English) Zbl 1073.11013
Let \(n \geq 4\), \(\beta_i = B_i/i\) where \(B_i\) is the \(i\)th Bernoulli number and let \(H_n = 1 + 1/2 + \cdots + 1/n\). In 1978 [J. Number Theory 10, 297–302 (1978; Zbl 0379.10007)], H. Miki obtained the identity \[ \sum_{i=2}^{n-2} \beta_i \beta_{n-i} - \sum_{i=2}^{n-2} {n \choose i} \beta_i \beta_{n-i} = 2H_n\beta_n \] by showing that the two sides were congruent modulo sufficiently large primes. The author gives a simple proof of a family of identities including this one by deriving in two ways expansions of Stirling numbers \(S(m+n, n)\) of the second kind in ascending powers of \(m\) and comparing coefficients. He indicates that one can prove a common generalization of this identity and one due to C. Faber and R. Pandharipande [Invent. Math. 139, 173–199 (2000; Zbl 0960.14031)]: \[ {n \over 2}\sum_{i=2}^{n-2} {{B_i({1 \over 2})}\over i} {{B_{n-i}({1 \over 2})}\over{n-i}} - \sum_{i=0}^{n-2} {n \choose i} B_i \biggl({1 \over 2}\biggr)\beta_{n-i} = H_{n-1}B_n\biggl({1 \over 2}\biggr) \] where \(B_n(\lambda)\) is a Bernoulli polynomial.

MSC:
11B68 Bernoulli and Euler numbers and polynomials
11B73 Bell and Stirling numbers
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[1] Dilcher, K., Sums of products of Bernoulli numbers, J. number theory, 60, 23-41, (1996) · Zbl 0863.11011
[2] Faber, C.; Pandharipande, R., Hodge integrals and gromov – witten theory, Invent. math., 139, 173-199, (2000) · Zbl 0960.14031
[3] Gessel, I.; Stanley, R.P., Stirling polynomials, J. combin. theory ser. A, 24, 24-33, (1978) · Zbl 0378.05006
[4] Huang, I.-C.; Huang, S.-Y., Bernoulli numbers and polynomials via residues, J. number theory, 76, 178-193, (1999) · Zbl 0940.11009
[5] Miki, H., A relation between Bernoulli numbers, J. number theory, 10, 297-302, (1978) · Zbl 0379.10007
[6] Shiratani, K.; Yokoyama, S., An application of p-adic convolutions, Mem. fac. sci. kyushu univ. ser. A, 36, 73-83, (1982) · Zbl 0507.12011
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