Sahih Muslim Book 6, Hadith Number 2555.

Chapter : Completing of the fast on behalf of the dead.

Ibn ‘Abbas (Allah be pleased with them) reported: A man came to the Apostle of Allah (may peace be upon him) and said: Messenger of Allah, my mother has died (in a state) that she had to observe fasts of a month (of Ramadan). Should I complete (them) on her behalf? thereupon he (the Holy Prophet) said: Would you not pay the debt if your mother had died (without paying it)? He said: Yes. He (the Holy Prophet) said: The debt of Allah deserves more that it should he paid.