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Zbl 1228.11025
PetojeviÄ‡, Aleksandar; Srivastava, H.M.
Computation of Euler's type sums of the products of Bernoulli numbers.
(English)
[J] Appl. Math. Lett. 22, No. 5, 796-801 (2009). ISSN 0893-9659

Summary: In this work, the authors present several formulas which compute the following Euler type and Dilcher type sums of the products of Bernoulli numbers $B\sb n$: $$\Omega\sp {(m)}\sb n := \sum\Sb j\sb 1+\dots+j\sb m=n\\ (j\sb 1,\dots,j\sb m\geq1)\endSb \left(\matrix2n\\ 2j\sb 1,\dots,2j\sb m\endmatrix\right)B\sb {2j\sb 1}\cdots B\sb {2j\sb m}$$ and $$\Delta\sp {(m)}\sb n := \sum\Sb j\sb 1+\dots+j\sb m=n\\ (j\sb 1,\dots,j\sb m\geq0)\endSb\left(\matrix2n\\ 2j\sb 1,\dots,2j\sb m\endmatrix\right)B\sb {2j\sb 1}\cdots B\sb {2j\sb m}$$ respectively, where $$\left(\matrix n\\ k\sb 1,\dots,k\sb m\endmatrix\right)={n!\over k\sb 1!\cdots k\sb m!}$$ denotes, as usual, the multinomial coefficient.
MSC 2000:
*11B68 Bernoulli numbers, etc.

Keywords: Bernoulli numbers; Bernoulli polynomials; sums of products; Euler's sums; Dilcher's sums; multinomial coefficients; integer sequences; generating functions; Cauchy product of power series; Stirling numbers of the first kind

Cited in: Zbl 1234.11022

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