On Tornheim type series

In this work, Tornheim type series T(s₁,s₂,s₃;x,y,z;c)=∑_{m=1}^{∞}∑_{n=1}^{∞}((e^{2πi(mx+ny+(mc+n)z)})/(m^{s₁}n^{s₂}(mc+n)^{s₃})) is defined and some functional equations are obtained for this series. The special cases of these functional equations give many of the results in the literature for the Tornheim and alternating Tornheim series. In addition, formulas for some infinite series are given as applications of the obtained functional equations.