  In [8], P. Lecomte conjectured the existence of a natural and projectively
equivariant quantization. In [1], M. Bordemann proved this existence using the
framework of Thomas-Whitehead connections. In [9], we gave a new proof of the
same theorem thanks to the Cartan connections. After these works there was no
explicit formula for the quantization. In this paper, we give this formula
using the formula in terms of Cartan connections given in [9]. This explicit
formula constitutes the generalization to any order of the formulae at second
and third orders soon published by Bouarroudj in [2] and [3].
