Analytic Properties of the Sum $B_{1}(h,k)$

TL;DR

This paper explores the properties and interconnections of the finite sum $B_{1}(h,k)$ with classical sums like Dedekind and Hardy sums, introducing new relations and properties using Fibonacci numbers and polynomial relations.

Contribution

It provides new properties of $B_{1}(h,k)$ and establishes its connections with well-known sums, expanding understanding of its analytic structure.

Findings

Derived new properties of $B_{1}(h,k)$ using Hardy and Dedekind sums.

Established relationships between $B_{1}(h,k)$ and sums like Dedekind, Hardy, and Simsek sums.

Introduced a novel property of $B_{1}(h,k)$ involving Fibonacci numbers and polynomial relations.

Abstract

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 $B_{1}(h,k)$.