Is it permissible for me to use zakat funds to buy books for my son or to pay his tuition fees, while his father is spending on his own studies and building a house?

It is not permissible for a mother to give her zakat money to her son to help him with his study expenses, because his father, if capable, is obligated to provide for him. Furthermore, the فقهاء (jurists) state that it is not permissible to give obligatory zakat to one's parents, no matter how high their lineage, nor to one's children, no matter how low their lineage.