We study two families of polynomials that play the same role in the Temperley– Lieb algebra of a Coxeter group as the Kazhdan–Lusztig and R–polynomials play in the Hecke algebra of the group. Our results include recursions, non– recursive formulas, symmetry properties and expressions for the constant term. We focus mainly on non–branching Coxeter graphs.

We study two families of polynomials that play the same role in the Temperley– Lieb algebra of a Coxeter group as the Kazhdan–Lusztig and R–polynomials play in the Hecke algebra of the group. Our results include recursions, non– recursive formulas, symmetry properties and expressions for the constant term. We focus mainly on non–branching Coxeter graphs.