If a Muslim misses days of Ramadan due to a legitimate Shar'i excuse and is unable to make them up before the onset of the following Ramadan, does he incur the expiation of feeding a poor person for each day he makes up after Ramadan?

The majority of jurists hold that whoever delays making up for missed days of Ramadan without a valid excuse must make up those days and feed one poor person for each day. However, if there is a valid excuse, then nothing is required of them. As for one who delayed making up the days due to ignorance of the impermissibility of delaying, they are not obligated to pay the fidyah (expiatory feeding) but only to make up the fasts. If there is not enough time to make up all the days before the next Ramadan, you may complete the fasting after Ramadan, and no expiation is required of you.