Convolution identities for Cauchy numbers

Abstract

Let $${c_{n} (n \geqq 0)}$$ be the n-th Cauchy number, defined by the generating function $${x/\ln(1 + x) = \sum^{\infty}_{n=0}{c_{n}x^{n}/n!}}$$ . We investigate an explicit expressions for $${{(c_{l} + c_{m})^{n} := \sum^{n}_{j=0}\left (\begin{array}{ll} n \\ j \end{array}\right ){c_{l+j}c_{m+n-j}}}}$$ with arbitrary fixed integers $${l, m \geqq 0}$$ . If l = m = 0, then $${(c_{0} + c_{0})^{n} = -n(n - 2) c_{n - 1} - (n - 1)c_{n}}$$ , which was obtained by Zhao. The corresponding expressions $${(B_{l} + B_{m})^{n}}$$ for Bernoulli numbers $${B_{n}}$$ have been studied by several authors.