Identities for a Special Finite Sum Related to the Dedekind Sums and Fibonacci Numbers

The origin of this article is to achieve original equations related to the special finite sum C(μ,β;1), which is connected with Dedekind, Hardy, Simsek, and many other finite sums. By using the analytic properties of this sum, many useful identities are established between the C(μ,β;1) sum and other well-known finite sums. Through the use of these identities, the reciprocity law of this sum is obtained. Furthermore, another reciprocity law of the sum C(μ,β;1) is presented for μ and β are particular Fibonacci numbers. This remarkable result establishes a connection between number theory and analysis.