Parabolic Kazhdan–Lusztig and RR-polynomials for Boolean elements in the symmetric group

The parabolic analogue of the Kazhdan–Lusztig and RR-polynomials has been introduced by Deodhar [V. Deodhar, On some geometric aspects of Bruhat orderings. II. The parabolic analogue of Kazhdan–Lusztig polynomials, J. Algebra 111 (1987) 483–506]. Answering a question of Brenti, we give closed combinatorial product formulae for the parabolic RR-polynomials and for the parabolic Kazhdan–Lusztig polynomials of type qq in the case where the indexing permutations are smaller than the top transposition in Bruhat order. These formulae are valid in complete generality on the parabolic subgroup WJWJ.