Third and Higher Order Convolution Identities for Cauchy Numbers

Takao Komatsu, Yilmaz Simsek

The $n$-th Cauchy number $c_nP(n\geq0)$ are defined by the generating function $x/\ln(1+x)=\sum^\infty_{n=0}c_nx^n/n!$. In this paper, we deal with formulae of the type \[ um_{ubstack{l_1+\dots+l_m=\mu l_1,\dots,l_m\geq0}}\frac{\mu!}{l_1!\dots l_m!}(c_{l_1}+\dots+c_{l_m})^n=a_0c_{n+\mu}+\dots+a_{m-1}c_{n+\mu-m+1}, \] where the $a_i$ are suitable rational numbers, the $c_i$ are Cauchy numbers and \[ (c_{l_1}+\dots+c_{l_m})^n:=um_{ubstack{k_1+\dots+k_m=n k_1,\dots,k_m\geq0}}\frac{n!}{k_1!\dots k_m!}c_{k_1}+l_1\dots c_{k_m}+l_m. \] In particular, we give explicit formulae for $m=3$ and $m=4$.