Is the death of a child born out of wedlock, after his mother abandoned him and left him in a public place, considered premeditated murder? And what is the jurisprudential ruling for both the man and the woman regarding their neglect of the child?

If a mother abandons her child born out of wedlock, and the child dies of hunger or cold, she is negligent and liable. The ruling on her action depends on the circumstances:

Quasi-intentional homicide ( شبه العمد): If the mother leaves the child in a place where safety is highly probable, but the child dies, her action is quasi-intentional homicide. Intentional homicide (القتل العمد): If the mother leaves the child in a place where death is highly probable (due to extreme heat, cold, or lack of passersby), and the child dies, her action is intentional homicide. Quasi-intentional homicide (when probabilities are equal): If the probabilities are equal, and neither safety nor death is highly probable, her action is quasi-intentional homicide.

If it is intentional homicide, then Qisas (retaliation) is due. If it is quasi-intentional homicide, then Diyya (blood money) is due from her 'Aqilah (male relatives on the father's side) and Kaffarah (expiation) is due from her. As for the fornicator who had no hand in exposing the child to death, he must repent from fornication.