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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
##### Keywords:
Bernoulli numbers; Stirling numbers
Full Text:
##### References:
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