Non-negativity properties of
explicit formula expressing the coefficients of the \tilde
as a linear combinations of those of the corresponding
R-polynomials is given.
Several systems of inequalities satisfied by the coefficients of
are shown and applied to obtain a lower bound for the coefficient of q2 in
Kazhdan-Lusztig polynomials. Finally we present a basis of the space of
of a given degree with respect to which both Kazhdan-Lusztig and
have some nice (conjectural) non negativity properties.