A Note on Hardy type Sums and Dedekind Sums

Abstract

In \cite{csc}, Cetin et al. defined a new special finite sum which isdenoted by $C_{1}(h,k)$. In this paper, with the help of two-term polynomialrelation, we will give the explicit values of the sum $C_{1}(h,k).$ We willsee that for the odd values of $h$ and $k$, this sum only depends on onevariable. After that we will give many properties of this sum andconnections with other well-known finite sums such as the Dedekind sums, theHardy sums and the Simsek sums $Y(h,k).$ By using the Fibonacci numbers andtwo-term polynomial relation, we will also give some relations for thesesums.