Ata (may Allah have mercy on him) says that he heard Sayyiduna Abu Hurayrah (may Allah be pleased with him) say, concerning a sick person who falls ill and does not fast Ramadan, then recovers and does not fast until another Ramadan reaches him: He should fast the Ramadan that is present, fast the other one afterwards, and feed a poor person for each night.
Ibrahim bin Nafi' al-Jallab narrated this hadith from Umar bin Musa bin Wajih, from al-Hakam, from Mujahid, from Abu Hurayrah as a saying of the Prophet (peace and blessings be upon him) (marfu'), and it is nothing; Ibrahim and Umar are both matruk (abandoned).
It is reported from Sayyiduna Abdullah bin Umar and Sayyiduna Abu Hurayrah (may Allah be pleased with them both), concerning one who does not recover until another Ramadan reaches him, that he should feed (the poor) and owes no make-up. Al-Hasan, Tawus and al-Nakha'i (may Allah have mercy on them) say that he must make it up but owes no expiation, and this is what we hold, because of the saying of Allah Almighty: "Then (fast) a number of other days." [Surah al-Baqarah: 184]
Hadith Referenceالسنن الكبرى للبيهقي / كتاب الصيام / 8213