Analytic series expansion of the overlap reduction function for gravitational wave search with pulsar timing arrays

In our previous paper \cite{PTA1} we derived a generic expression for the pulse redshift the main observable for the Pulsar Timing Array (PTA) experiment for detection of gravitational waves for all possible polarizations induced by modifications of general relativity (GR). In this work we provide a generic expression of the overlap reduction function for PTA without using the short wavelength approximation for tensorial polarization. We are convinced, that the short wavelength approximation is not applicable to the overlap reduction function for PTA's, since the removal of the exponential terms in the integrand would lead to poles for $x, y$ and $l$ polarizations and discontinuities for $+$ and $\times$. In this work we provide a series expansion to calculate the integral exactly and investigate the behaviour of the series for short wavelength values via numerical evaluation of the analytical series. We find a disagreement for the limit of co-located pulsars with the Hellings \& Downs curve.