Third and higher order convolution identities for Cauchy numbers

Abstract

The $n$-th Cauchy number $c_n$ ($n\ge 0$) are defined by the generating function $x/\ln(1+x)=\sum_{n=0}^\infty c_n x^n/n!$.
In this paper, we deal with formulae of the type
$$
\sum_{l_1+\cdots+l_m=\mu\atop l_1,\dots,l_m\ge 0}\frac{\mu!}{l_1!\cdots l_m!}(c_{l_1}+\cdots+c_{l_m})^n=a_0 c_{n+\mu}+\cdots+a_{m-1}c_{n+\mu-m+1}\,,
$$
where the $a_i$ are suitable rational numbers, the $c_i$ are Cauchy numbers and
$$
(c_{l_1}+\cdots+c_{l_m})^n:=\sum_{k_1+\cdots+k_m=n\atop k_1,\dots,k_m\ge 0}\frac{n!}{k_1!\cdots k_m!}c_{k_1+l_1}\cdots c_{k_m+l_m}\,.  
$$
In special, we give explicit formulae for $m=3$ and $m=4$.