Let (W,S) be a Coxeter system and write a31b3ef57e178101b8345074a823b" title="Click to view the MathML source">PW(q) for its Poincaré series. Lusztig has shown that the quotient af978c0ac66f54985d36517c" title="Click to view the MathML source">PW(q2)/PW(q) is equal to a certain power series e87d356a19745d42618e1f45ed09a" title="Click to view the MathML source">LW(q), defined by specializing one variable in the generating function recording the lengths and absolute lengths of the involutions in W . The simplest inductive method of proving this result for finite Coxeter groups suggests a natural bivariate generalization depending on a subset J⊂S. This new power series specializes to e87d356a19745d42618e1f45ed09a" title="Click to view the MathML source">LW(q) when e8ddf5285a71e821c9" title="Click to view the MathML source">s=−1 and is given explicitly by a sum of rational functions over the involutions which are minimal length representatives of the double cosets of the parabolic subgroup e8ed3d40215f869a" title="Click to view the MathML source">WJ in W. When W is an affine Weyl group, we consider the renormalized power series with J given by the generating set of the corresponding finite Weyl group. We show that when W is an affine Weyl group of type A , the power series a357496b3d64a3a874aa9" title="Click to view the MathML source">TW(s,q) is actually a polynomial in s and q with nonnegative coefficients, which turns out to be a q-analogue recently studied by Cigler of the Chebyshev polynomials of the first kind, arising in a completely different context.