Estimating additive character sums for Fuchsian groups

In the usual construction of non-holomorphic Eisenstein series, for a general Fuchsian group, a multiplicative character may be included. The properties of these series are well known. Here we instead include an additive character and develop the properties of the resulting series. We pay particular attention to additive characters that are non-cuspidal, i.e., that are not zero on some parabolic generators. These series may be used to estimate certain additive character sums. For example, asymptotics for a weighted sum over group elements that counts the number of appearances of a fixed generator of the Fuchsian group are obtained.