Before i begin translating this article, please be informed that the scholars of Islam have differed upon this issue. There are more than 3 views attributed to 4 major group of scholars of the past and present.
Imam Abu `Esa at-tirmidhi said in his Jami`:
“The people of knowledge differ on this issue. Some of them say a person is to fast on behalf of the deceased and this is the view of Aḥmad and Ishaaq. They said when the deceased made a vow to fast then someone is to fast on their behalf and if he has to make up a fast from Ramaḍan then a poor person is to be fed on his behalf. Mālik, Sufyān and Shāfiʿī said, “No one can fast on anyones behalf”
So should you follow another opinion, its fine. However i request capable Muslims to research into this issue and follow that which you feel compliments the view closest to the evidence. In the below article , shaykh zubair r.h also took the opinion of one group of scholars and he believed this view to be the most authentic one.
Question : May allah facilitate you with a long life to preach qur’an and sunnah and defend allah’s religion. Ameen. Respected Shaykh (Zubair `Ali Zai ) I had heard my Ustadh, Shaykh Ul-Qur’an wal-Hadeeth `Allamah `Abdus-salaam Rustami say that
“ when someone who has pending Ramadan fasts for him , dies then his inheritors or guardians can fast on his behalf and if someone wants to give fidyah (amount or food) on behalf of the deceased then it is permissible too. The stance of fasting on behalf of the deceased is proven from a marfoo` hadeeth whereas the stance of paying fidya is proven form mawqoof “
But Shaykhul-Hadeeth Mawlana Gawhar Rahmaan rahimahullah in his Fatawa Tafheem al-Masail Volume 1, Page 142 have used two ahadeeth to state that “No one can pray on behalf of the deceased nor can they fast on their behalf”

The narrations used by Shaykh `Abdus salaam to deem both fasting and fidya permissible are :

1) `Aisha radhiallahu `anha narrated that the Messenger of Allah sallalahu `alayhi wa sallam said that “ whoever dies and has (pending) fasts let his guardian fast for him” [Ref: Saheeh Bukhaari 1/262, Saheeh Muslim 1/362]
2) Ibn `Abbas radhiallahu `anhu narrated that one day a man came to the Messenger of Allah sallalahu `alayhi wa sallam and said “ Oh Messenger of Allah (sallalhu `alayhi wa sallam ) My mother has died and she has a month of fast pendent , should I fulfil it for her? The messenger of Allah said “ if your mother had an debt to be paid wouldn’t you have paid for her?” He replied “ Yes for sure” So he sallalahu `alayhi wa sallam said “ Allah’s debt is more deserving to be paid” [Ref: Saheeh Muslim 1/362]

The narrations used by Shaykh Gawhar rahman rahimahullah to negate fasting and fidyah are :

a) Ibn `Abbas radhiallahu `anhu narrated that the Messenger of Allah sallalahu `alayhi wa sallam said “No one is to pray on behalf of another and no one is to fast on behalf of another but may feed on his behalf [Ref: Sunan al-Kubra lil-Bayhaqi (4/257); Mushkilul-Athar (3/141); Nasbur-raya by Imam Zayli (2/463) and Ibn Hajr said the sanad is saheeh ad-diraya (page 177)]
b) Ibn `Umar radhiallahu `anhu said that the Messenger of Allah sallalahu `alayhi wa sallam said “ No one is to pray on another’s behalf, nor is anyone to fast on someone’s behalf but he may give charity and sacrifice for him (or his behalf)” [Ref: Sunan Kubra (4/257) Zayli (Ibid); Mushkilul-Athar (Ibid); Ibn hajar authenticated it (Ibid)]

Answer by Hafidh Zubair `Ali Zai (rahimahullah) :

`Aisha radhiallahu `anha narrated that the Messenger of Allah sallalahu `alayhi wa sallam said “ Whoever dies and has fasts upon him, let his guardian fast for him” [Ref: Saheeh Bukhaari 1952; Saheeh Muslim]
It should be noted that here the phrase “Whoever dies and has fasts upon him” is for the fasts of vow only and the following evidences suggests so :
1) Ibn `Abbas radhiallahu `anhu narrates that a woman came to the Messenger of Allah sallalahu `alayhi wa sallam and said “ Oh Messenger of Allah my mother died while having vowed to fast , so should I fulfil it on her behalf?”. The Messenger of Allah sallalahu `alayhi wa sallam said “ Do you think that if your mother owed a debt and if you were to pay it on her behalf would it suffice her debt?” She replied “ Yes” So he sallalahu `alayhi wa sallam replied “ So fast on her behalf”
[Ref: Saheeh Muslim 1148 Darrussalaam edition TN (translators note): Even imam Ibn Hazam zahiri rahimahullah authenticated this as mentioned by shaykh albani in Ahkam ul-Janaiz 215]
2) Another incident is wherein a woman during her sea travel vowed to fast a month but she died before she could fast so her sister informed the Messenger of Allah sallalahu `alayhi wa sallam about this incident so he sallalahu `alayhi wa sallam asked her to fast on her sister’s behalf
[Ref: Sunan as-sughra li-Nasa’e, Kitab al-Emaan wan-nuzur under the chapter “About one who vows to fast but dies before fulfilling it “ (7/20, H:3847) with an authentic chain and Ibn Khuzayma authenticated it (2054)]
Imam Abu Dawood brought a similar narration in his Sunan (that mentiones the women vowed that if Allah saved her, she would fast one month Hadeeth: 3308 , authentic) and put it under the chapter “ ،ھذا فی النذر ، وھو قول أ حمد بن حنبل . | Chapter of fulfilling the vows of the deceased”
Since one hadeeth explains another the above narrations explain that the fulfillment of fasts is in regards to vowed fasts only
3) The Narrator of the first hadeeth (no. 1 )mentioned above Sayyidah (Ummal Mu’mineen – may allah disgrace in both the worlds those who disgrace her in any way whatsoever ) `Aisha radhiallahu `anha was asked by a woman “My mother passed away and she has the fast of Ramadan pending , should I fast on her behalf?” `Aisha ( the wife of the best of allah’s creation in this world and hereafter ) said “ No but rather feed a poor person for every day and this is better for her than your fasting “
[Ref: Sharh Mushkil ul-Athar Tahawi (6/178) with an authentic chain and Turkmanai Hanafi authenticated it in al-Jawhar (4/257) | ان امھا ماتت ولیھا من رمضان، فقالت لعا ئشۃ : أقضیہ عنھا؟ قالت : لا، بل تصد قی عنھا وکان کل یوم نصف صاع علی کل مسکین . ]
From the fatwa of the narrator of the same hadeeth i.e. `Aisha it is evident that fasting on behalf of someone’s regular missed fasts is not to be done rather food is fed but fasting is observed only for vowed fasts of the deceased
4) Ibn `Abbas (the narrator of the second hadeeth given by shaykh `abdus salaam no. 2 above) radhiallahu `anhu said “No one is to pray on behalf on another and no one is to fast on behalf on another but instead let him feed on his behalf against each day (of missed fast) one mudd of wheat “
[Ref: Sunan al-Kubra of Nasa’ee (2/175, H :2178) with an authentic chain, and also authenticated by Hafidh Ibn Hajar in Diraya (1/283; H:375); TN : Another narration in Ahkamul-Janaiz of Shaykh al-Albani mentions a similar narration from Ibn `Abbas wherein he said whoever dies having fasts of Ramadan pending then let him give food in its exchange ,but if he had made a vow then his wali should fast on his behalf]
5) Imam (of the `Aqeedah of Ahlus sunnah – who stood like a wall against the innovations of his time) Ahmad ibn Hanbal and Imam Ishaaq ibn Rahawy both said “If the deceased dies having vowed to fast then let (his wali) fast on his behalf and if he has missed fasts of Ramadan then feed on his behalf “ [Ref: Sunan tirmidhi 718, Masail Imam Ahmad wa Ishaaq via Ishaq ibn Mansoor (1/288 ;679 or page 96) Sunan al-Kubra lin-Nasa’ee (2917)]
Imam Abul `Abbas Ahmad ibn `Amr ibn Ibrahim al-Qurutibi (D. 656 A.H) said
“And this is the statement of Ahmad , (Imam Ahlus sunnah) Layth ibn Sa`d al-Misri ; (Imam) Abu `Ubayd except that they specified this hadeeth (of making up fasts) to be for vowed fasts “ [Ref: Al-Mafhoom (3/208 , under H :1014)]
It is evident from both the narrators of the first two hadeeth (mentioned by Shaykh `Abdus salaam) that one is to fast on behalf of the deceased only if the deceased had vowed fasts pending but for missed Ramadan fasts then fidyah is to be given instead of fasting. This is the understanding of the narrators (`Aisha and Ibn `Abbas ) and the salafus saliheen .
As for Mawlana Gawhar rahman mentioning the hadeeth of Ibn `Abbas via the prophet from Mushkilul Athar, Bayhaqi, Nasb ur-raya and dirayah then it is to be noted that in these references the hadeeth wording comes from Ibn `Abbas as his fatwa and not from the Messenger of Allah so this shouldn’t be attributed to the Prophet . Imam Zayli Hanafi (rahimahullah) also said “Ghareeb Marfoo`an” that it is strange/odd from a marfoo` way. [Ref: Nasbur raya 2/463]
You have mentioned both the narrations from Gawhar rahman sahib from his Tafheem masail part 1 and as per my copy (Published by jinnori 1993 edition) I have them on page 123-124 and he has erred in deeming what was mawqoof to be marfoo`. In reality they are not from the prophet but the fatawa of the sahaba.
Gawhar khan sahib narrated both those narrations as marfoo` but he has erred and they are not marfoo`. More so, `Abdur razzaq and others have recorded this from Ibn `Umar radhiallahu `anhu as his statement (fatwa) [Ref: Nasbur-raya 2/463]
Hafidh ibn Hajar said “ this is saheeh from Ibn `Umar mawqoofan “( i.e. from the statement of a sahaabi and not the prophet) [Ref: Daraya 1/283 H:375]
Note that due to the tadlees of `Abdur razzaq, the Ibn `Umar narration in his Musannaf is weak. On the contrary Imam Bayhaqi (al-`Ashari) rahimahullah narrated from Ibn `Umar radhiallahu `anhu saying “No one is to fast for another but give charity ( i.e feed) from (his) wealth to poor for every day(‘s missed fast) “ [Ref: Sunan Kubra 4/254, Authentic]
The chain of this narration is saheeh and Imam Malik narrated this without a chain with some additions in his Muwatta [Ref: 1/303, H: 671, Dha`eef – Muwatta from the (famous) route of Yahya ibn Yahya]
Imam Abul Jahm al-`Ala ibn Moosa Ibn `Atiy al-Baghdadi rahimahullah said “ I heard Layth ibn Sa`eed from Nafi` from Ibn `Umar who said “No one should fast on another’s behalf and no one should perform Hajj on another’s behalf” [Ref: Juzz Abil-Jahm 24, with an authentic chain]
In this statement “ no one should perform hajj on another’s behalf” is disputed (or odd) because it is established that the prophet allowed doing Hajj on someone else’s behalf provided the one who doing it performed hajj for himself first (once in his life time)
Imam Ibn `Abdul Birr has quoted an Ijmaa` that no one is to pray on anyone’s behalf [Ref: At-tamheed 9/133]
[Ref: Translated from Shaykh Zubair `Ali Zai’s Article in al-Hadeeth no. 87, Page 9 onwards. Slightly abbreviated as and when needed]