Non-negativity properties of R-polynomials

An explicit formula expressing the coefficients of the tilde R-polynomials as a linear combinations of those of the corresponding R-polynomials is given. Several systems of inequalities satisfied by the coefficients of the R-polynomials 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 polynomials of a given degree with respect to which both Kazhdan-Lusztig and R-polynomials have some nice (conjectural) non negativity properties.