Abstract
We investigate sums of products of Cauchy numbers including poly-Cauchy numbers: Tm(k)(n)=∑i1+⋯+im=n, i1,…,im≥0(ni1,…,im)ci1⋯cim-1 cim(k) (m≥1,n≥0). A relation among these sums Tmkn 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.
Highlights
The Cauchy numbers cn are defined by the integral of the falling factorial: cn ∫ x (x − 1) ⋅⋅ ⋅
We investigate sums of products of Cauchy numbers including poly-Cauchy numbers: Tm(k)(n)
We 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
Summary
We investigate sums of products of Cauchy numbers including poly-Cauchy numbers: Tm(k)(n) We 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. Are sometimes called the Bernoulli numbers of the second kind (see e.g., [2, 3]). In 1997 Kaneko [9] introduced the poly-Bernoulli numbers Bn(k) (n ≥ 0, k ≥ 1) by the generating function
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