  The aim of this paper is to construct new Dedekind type sums. We construct
generating functions of Barnes' type multiple Frobenius-Euler 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 Frobenius-Euler functions, we also define analogues of
Hardy-Berndt type sums. We give some new relations related to to these sums as
well.
