What is the ruling regarding someone who passed away having not fasted two months of Ramadan due to illness and old age, and is it incumbent upon his children to fast on his behalf or to offer expiation (fidyah)?

Sons and daughters do not fast on behalf of their deceased father who was exempt from fasting due to old age. Rather, they should pay a fidyah (expiation) on his behalf, amounting to one mudd (a measure of volume) of the prevalent local food for each day, because it was an obligation upon him during his lifetime that he did not fulfill. This is the sounder of the two opinions, and its evidence is the recitation of Ibn Abbas, may Allah be pleased with him, of the verse: ﴿وَعَلَى الَّذِينَ يُطَوَّقُونَهُ فِدْيَةٌ طَعَامُ مِسْكِينٍ﴾ [Al-Baqarah: 184], which means those who are burdened with fasting but are unable to perform it. This has been narrated from Ibn Umar, Ibn Abbas, Anas, and Abu Hurairah.