Convolution identities for Cauchy numbers (Q485013)

Let \(c_n\) \((n\geq 0)\) be the \(n\)th Cauchy number, defined by the generating function \(x/ \ln (1 + x)= \sum_{n=0}^{\infty} c_n x^{n}/n!\). With the classical umbral calculus notation, define \((c_l + c_m)^{n}\) for \( l, m, n\geq 0 \) by \((c_l + c_m)^{n} :=\sum_{j=0}^{n} {{n} \choose{j}} c_{l+j}c_{m+n-j}\). In this paper, an explicit formula for \((c_l + c_m)^{n}\) (\(n\geq 0\)) together with some initial cases are given. Note that if \(l=m=0\), then \((c_0 + c_0)^{n}=-n(n-2)c_{n-1}-(n-1)c_n\), which was obtained by \textit{F.-Z. Zhao} [Discrete Math. 309, No. 12, 3830--3842 (2009; Zbl 1191.05008)] and the corresponding expressions \((B_l + B_m)^{n}\) for Bernoulli numbers \(B_n\) have been studied by several authors (\(c_n=n!B_n\) holds).