Factorization formulas for Macdonald polynomials

The aim of this note is to give some factorization formulas for different versions of the Macdonald polynomials when the parameter t is specialized at roots of unity, generalizing those given in [A. Lascoux, B. Leclerc, J.-Y. Thibon, Green polynomials and Hall–Littlewood functions at roots of unity, European Journal of Combinatorics (15) (1994) 173–180 [8]; A. Lascoux, B. Leclerc, J.-Y. Thibon, Fonctions de Hall–Littlewood et Polynomes de Kostka-Foulkes aux racines de l’unité, Comptes Rendus de l’Academie des Sciences, Paris 316 (1993) 1–6] for Hall–Littlewood functions.