It is a bit difficult to tell what the Rexroth formula is exactly without seeing their definitions. I believe the two formulas are the same, just with the Rexroth version using some slightly different terminology.
They use Eoil which I presume is a measure of the stored energy in the oil, and then substitute H/2 x AR to denote the volume. I take this to mean some kind of rod area by perhaps a height? It is not clear from context. In any case, it appears to arrive at the same expression, just with slightly different symbols for the various terms.
I recommend this thread, Computing the Natural Frequency of the Cylinder and Load, to see a worked out example.