Special functions related to Dedekind-type DC-sums and their applications

In this paper, we construct trigonometric functions in the form of a sum T _p (h, k) which is referred to as a Dedekind-type DC-(Dahee and Changhee) sum. We establish analytic properties of this sum, find its trigonometric representations, and prove a reciprocity theorem for these sums. Furthermore, we obtain relationships between the Clausen functions, polylogarithm function, Hurwitz zeta function, generalized Lambert series (G-series), Hardy-Berndt sums, and the sum T _p (h, k). We also give some applications related to these sums and functions.