Special Functions Related to Dedekind Type DC-Sums and their Applications

TL;DR

This paper introduces and analyzes Dedekind type DC-sums, exploring their properties, representations, and relations to special functions, with applications in mathematical analysis and number theory.

Contribution

It constructs and studies the properties of Dedekind type DC-sums, including their trigonometric representations and reciprocity, and establishes connections with various special functions.

Findings

Derived trigonometric representations of the sums

Proved reciprocity theorem for the sums

Established relations with Clausen, Polylogarithm, and Hurwitz zeta functions

Abstract

In this paper we construct trigonometric functions of the sum T_{p}(h,k), which is called Dedekind type DC-(Dahee and Changhee) sums. We establish analytic properties of this sum. We find trigonometric representations of this sum. We prove reciprocity theorem of this sums. Furthermore, we obtain relations 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.