Twisted Dedekind Type Sums Associated with Barnes’ Type Multiple Frobenius-Euler l-Functions

The aim of this paper is to construct new Dedekind type sums. We construct generating functions of Barnes’ type multiple FrobeniusEuler numbers and polynomials. By applying Mellin transformation to these functions, we define Barnes’ type multiple l-functions, which interpolate Frobenius-Euler numbers at negative integers. By using generalizations of the Frobenius-Euler functions, we define generalized Dedekind type sums and prove corresponding reciprocity law. We also give twisted versions of the Frobenius-Euler polynomials and new Dedekind type sums and corresponding reciprocity law. Furthermore, by using p-adic q-Volkenborn integral and twisted (h, q)-Bernoulli functions, we construct p-adic (h, q)-higher order Dedekind type sums. By using relation between Bernoulli and FrobeniusEuler functions, we also define analogues of Hardy-Berndt type sums. We give some new relations related to to these sums as well.