  The Cauchy-type product of two arithmetic functions $f$ and $g$ on
nonnegative integers is defined as $(f\bullet g)(k):=\sum_{m=0}^{k} {k\choose
m}f(m)g(k-m)$. We explore some algebraic properties of the aforementioned
convolution, which is a fundamental-characteristic of the identities involving
the Bernoulli numbers, the Bernoulli polynomials, the power sums, the sums of
products, henceforth.
