Non-negativity properties of -polynomials

An explicit formula expressing the coefficients of the 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 q 2 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.