  We study two families of polynomials that play the same role, in the
generalized Temperley Lieb algebra of a Coxeter group, as the Kazhdan Lusztig
and R polynomials in the Hecke algebra of the group. Our results include
recursions, closed formulas, and other combinatorial properties for these
polynomials. We focus mainly on non branching Coxeter graphs.
