Is it permissible to buy one's mother's gold at half its value, and is it permissible for her to gift gold to one of her children without the others on an occasion specific to that child?

It is permissible to sell an item for less than its original price if its owner is not legally incapacitated, such as a mother selling gold to her daughter for less than its price, unless there is an obvious favoritism towards other children without a legitimate Sharia reason. As for a parent giving something to one of their children for an occasion like marriage or illness, it is permissible, and they should give a similar gift to their other children on similar occasions.