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.

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