The Prophet ﷺ as saying: If anyone dies when some fast is due from him (i.e. which he could not keep) his heir must fast on his behalf. Abu Dawud said: This applies to the fast which a man vows ; and this is the opinion of Ahmad b. Hanbal.  

أَنَّ النَّبِيَّ ﷺ قَالَ مَنْ مَاتَ وَعَلَيْهِ صِيَامٌ صَامَ عَنْهُ وَلِيُّهُ  

If a man falls ill during Ramadan and he dies, while he could not keep the fast, food will be provided (for the poor men) on his behalf ; there is no atonement (for his fasts) due from him. If there is some vow which he could not fulfill, his heir must atone on his behalf.  

إِذَا مَرِضَ الرَّجُلُ فِي رَمَضَانَ ثُمَّ مَاتَ وَلَمْ يَصُمْ أُطْعِمَ عَنْهُ وَلَمْ يَكُنْ عَلَيْهِ قَضَاءٌ وَإِنْ كَانَ عَلَيْهِ نَذْرٌ قَضَى عَنْهُ وَلِيُّهُ