We give explicit formulas for several classes of Kazhdan-Lusztig polynomials of the symmetric group which are related to others already considered in the literature. In particular, we generalize two theorems of Brenti and Simion [Explicit formulae for some classes of Kazhdan-Lusztig polynomials, J. Algebraic Combin. 11 (2000),  187-196],  we prove an unpublished conjecture of Brenti, and we treat the case missing in Theorem 4.8 of [F. Caselli, Proof of two conjectures of Brenti and Simion on Kazhdan-Lusztig polynomials,  J. Algebraic Combin. 18 (2003),  171-187].