If you're referring to the original PageRank algorithm then the solution of the linear equations (for d<1; d = damping factor) is independent of the initial vector. Moreover, the solution is unique and even normalization doesn't play a role. (Normalization would only be important for the case d=1 which corresponds to the calculation of eigen vectors.) In practice one is using the vector of the last calculation to speed up the calculations.
Even in the TrustRank model with artificial sources initial conditions don't play a role.
There is no recursively definition. Of course, iteration schemes to calculate the solution of the linear equation are recursive. However, the definition of PR isn't recursive.
By the way, in principle you could also calculated PR within one step without initial guess as already mentioned here: Google meets Heisenberg [webmasterworld.com]. Of course, in practice this wouldn't work for such a large system.