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

The parabolic analogue of the Kazhdan–Lusztig and R -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 R -polynomials and for the parabolic Kazhdan–Lusztig polynomials of type q 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 W J .