  In \cite{csc}, Cetin et al. defined a new special finite sum which is denoted
by $B_{1}(h,k)$. In this paper, with the help of the Hardy and Dedekind sums we
will give many properties of the sum $B_{1}(h,k).$ Then we will give the
connections of this sum with the other well-known finite sums such as the
Dedekind sums, the Hardy sums, the Simsek sums $Y(h,k)$ and the sum
$C_{1}(h,k)$. By using the Fibonacci numbers and two-term polynomial relation,
we will also give a new property of the sum $B_{1}(h,k)$.
