Must the firstborn, who is the result of an illegitimate relationship and carries my name, retain my name, or should I make him carry his mother’s name only?

The child resulting from an illegitimate relationship does not have his lineage affixed to that of the biological father; he neither inherits from him nor is inherited by him. The legitimate children of the father are not his siblings. Rather, the child is attributed to his mother, and she inherits from him and he inherits from her. Because this child results from an illegitimate relationship, his lineage from the father's side is severed, and thus he is not permitted to bear his name.