A study on a type of poly-Dedekind type DC sums

As a new generalization of the Dedekind type DC sums, the poly-Dedekind type DC sums are introduced, which are obtained from the Dedekind type DC sums by replacing Euler functions by poly-Euler functions of arbitrary indices arising from polylogarithm functions. By using the definitions of polylogarithm functions, Stirling numbers of second kind, and Euler polynomials, several properties of poly-Euler functions are obtained, including the relations between this type of poly-Euler numbers and Stirling numbers of second kind, the relations between this type of poly-Euler polynomials and Euler polynomials and Stirling numbers of second kind. The poly-Dedekind type DC sums are shown to satisfy a reciprocity relation. It can be further explored in connection with modular forms, ζ functions, and trigonometric sums, just as in the cases of Apostol-Dedekind sums.