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Sums of products of Cauchy numbers, including poly-Cauchy numbers. (English) Zbl 1294.11027

Summary: We investigate sums of products of Cauchy numbers including poly-Cauchy numbers: \[ T_m^{(k)}(n)=\sum_{\substack{ i_1+\cdots+i_m=n,\\i_1,\ldots,i_m\geq 0 }} {n \choose i_1,\ldots, i_m} c_{i_1}\cdots c_{i_{m-1}}c_{i_m}^{(k)}, \quad (m\geq 1,\, n \geq 0) . \]
A relation among these sums \(T_m^{(k)}(n)\) shown in the paper and explicit expressions of sums of two and three products (the case of \(m=2\) and that of \(m=3\) described in the paper) are given. We also study the other three types of sums of products related to the Cauchy numbers of both kinds and the poly-Cauchy numbers of both kinds.

MSC:

11B75 Other combinatorial number theory
11B68 Bernoulli and Euler numbers and polynomials
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References:

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