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In our previous work \cite{PTA2} we calculated the overlap reduction functionfor the tensor polarization without employing the short wavelengthapproximation, this was done by obtaining a power series of nested sums whichis valid for all gravitational wave frequencies and pulsar distances. In thiswork we generalize the power-series expansion method to vector and scalarpolarizations. We have compared our expression for the breathing and vectormodes with previous literature. We present for the first time analyticexpressions for the overlap reduction function of the longitudinal mode for allangles between the pulsar pairs.

Series expansion of the overlap reduction function for scalar and vector polarizations for gravitational wave search with pulsar timing arrays