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We know Quran is a concise book, not a single word there is without reason.
Every single word is decorated and managed like a pearl in necklace, and not a single word wasted but when you see the Surah Rahman a single ayah "Fabe-aiye alae ..." repeated 31 times.
What could be the reason of using it again and again?
As said in my comment repetition in Arabic language is a way to emphasize things.
On the repeated statement itself
Before addressing this with details and quotes from scholars I'd like to point at two ahadith ibn Kathir has mentioned in his tafsir (see for example in qtafsir) in the introduction of the surah:
Jabir [may Allah be pleased with him] said: “The Messenger of Allah came out to his Companions, and recited Surat Ar-Rahman from its beginning to its end for them, and they were silent. So he said: ‘I recited it to the Jinns on the ‘Night of the Jinns,’ and they had a better response to it than you did. Each time I came to Allah’s saying: ‘Which of your Lords favor do you deny.’ They said: “We do not deny any of Your favors our Lord! And Yours is praise.” (Jami' at-Tirmidhi, ibn Kathir said it was also compiled by al-Bazzar)
Abu Ja`far Ibn Jarir recorded that 'Abdullah bin 'Umar said that the Messenger of Allah recited Surat Ar-Rahman, or it was recited before him, and he said,
(Why do I hear the Jinn giving a better response to their Lord than you) They said, "Why is that, O Allah's Messenger" He said,
(Whenever I recited the statement of Allah the Exalted, (Then which of the blessings of your Lord will you both deny)( The Jinn responded by saying, "None of the favors of our Lord do we deny.'')" Al-Hafiz Al-Bazzar also collected this Hadith. (Source qtafsir)
Note that the dual plural used in these verses "فبأي آلاء ربكما تكذبان" is referring to the two species or creation of Allah who are addressed by the words of the Qur'an: humans and Jinn. In surat an-Najm (53:55) when Allah addressed the prophet () -only- the Qur'an used the singular:
فَبِأَيِّ آلَاءِ رَبِّكَ تَتَمَارَىٰ
On pseudo scientific numerical interpretations
You may find some pseudo scientific proofs saying that this is related to the number 7 and creating a relation between the 31 repetitions and some picked verses of the surah as the first verse in which this is quoted is verse 13 and the letters of this verses are each time 18 (13+18=31) and the number which is generated when placing the verse numbers in a raw is a multiple of 7 etc.
Note that this applies only for the Kufi count (See also my answer on How many ayaths are there in Al-Quran?) which is used in Kufi qiraa't of the Qur'an such as Hafs 'an 'Assim as they divide the first verse of the surah into two as for the majority (all other counts) the verses:
The Most Merciful, Taught the Qur'an (55:1-2)
actually is one single ayah (verse) so the scholars have different opinions about the number of verses in this surah between 76-78 verses (al-Qurtobi quoted it is 76 -see here in Arabic- while the Kufi count is 78).
On repetitions in the Qur'an generally
The main reason of repetitions in the Qur'an is to emphasize a meaning to enforce it and to confirm it.
The Qur'an actually includes different types of repetitions:
The scholars said about the repetitions in the Qur'an (Source fatwa islamqa #82856):
Al-Suyooti (may Allaah have mercy on him) said:
There are a number of reasons for repetition:This includes confirmation. It is said that if words are repeated the meaning is confirmed. Allaah has indicated the reason why He repeats stories and reminders in the Qur’aan, as He says (interpretation of the meaning):
“and have explained therein in detail the warnings, in order that they may fear Allaah, or that it may cause them to have a lesson from it” [Ta-Ha 20:113]
It also includes emphasis.
It also includes highlighting a point so that the words will be accepted, such as the verse:
“And the man who believed said: ‘O my people! Follow me, I will guide you to the way of right conduct [i.e. guide you to Allaah’s religion of Islamic Monotheism with which Moosa (Moses) has been sent]. (39) "O my people! Truly, this life of the world is nothing but a (quick passing) enjoyment” [Ghaafir 40:38].In the passage the call is repeated for this reason.
Another example is if the speech is lengthy and there is the fear that what was said at the beginning may be forgotten, so it is repeated a second time to keep the idea fresh. For example, in the verses:
“Then, verily, your Lord __ for those who do evil (commit sins and are disobedient to Allaah) in ignorance and afterward repent and do righteous deeds, verily, your Lord thereafter, (to such) is Oft‑Forgiving, Most Merciful” [al-Nahl 16:119]
“Then, verily, your Lord _for those who emigrated after they had been put to trials and thereafter strove hard and fought (for the Cause of Allaah) and were patient, verily, your Lord afterward” [al-Nahl 16:110]
“And when there came to them (the Jews), a Book (this Qur’aan) from Allaah confirming what is with them [the Tawraat (Torah) and the Injeel (Gospel)], although aforetime they had invoked Allaah (for coming of Muhammad صلى الله عليه وسلم) in order to gain victory over those who disbelieved, then when there came to them that which they had recognised, they disbelieved in it” [al-Baqarah 2:89]
“Think not that those who rejoice in what they have done (or brought about), and love to be praised for what they have not done,— think not you that they are rescued from the torment” [Aal ‘Imraan 3:188]
“Verily, I saw (in a dream) eleven stars and the sun and the moon — I saw them” [Yoosuf 12:4]
It may also be done for emphasis, as in the verses:
“The Inevitable (i.e. the Day of Resurrection)! What is the Inevitable?” [al-Haaqqah 69:1]
“Al‑Qaari‘ah (the striking Hour, i.e. the Day of Resurrection). What is the striking (Hour)?” [al-Qaari’ah 101:1]
“And those on the Right Hand —how (fortunate) will be those on the Right Hand?” [al-Waaqi’ah 56:27]
al-Itqaan fi ‘Uloom al-Qur’aan (3/281, 282).
Al-Suyooti (may Allaah have mercy on him) said:
Repetition is more eloquent than merely emphasizing and it is reflective of good style, contrary to what some people think.
Al-Itqaan fi ‘Uloom al-Qur’aan (3/280), Mu’sasat al-Nada’ edition.
Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him) said:
There is no pointless repetition in the Qur’aan, rather there are benefits in every repetition.
Majmoo’ al-Fataawa (14/408)
Some comments and interpretations of the repeated statement from tafsir books
As for our example ibn 'Ashur in his at-Tahrir wa tanwir -see here in Arabic- in the introduction of the surah when quoting the goals and subjects of the surah said:
In the following I will translate here and later from Arabic, these are my own translation take it into account
ومن بديع أسلوبها افتتاحها الباهر باسمه الرحمان وهي السورة الوحيدة المفتتحة باسم من أسماء الله لم يتقدمه غيره .
Part of the exquisite eloquence in this surah is starting it by Allah's name ar-Rahmaan. This surah is the only one that has been started by a name of Allah here no other word was placed before!ومنه التعداد في مقام الامتنان والتعظيم بقوله فبأي آلاء ربكما تكذبان إذ تكرر فيها إحدى وثلاثين مرة وذلك أسلوب عربي جليل كما سنبينه .
And among its eloquence is the enumeration of the dignity of the expression of gratitude and glorification by "So which of the favors of your Lord would you deny?" as it was repeated 31 times and this is a great Arabic elegance of style which we will show (later)
Imam al-Qurtobi before showing examples of use of repetition from Arabic poetry said about this verse -see here in Arabic:
فالتكرير في هذه الآيات للتأكيد والمبالغة في التقرير ، واتخاذ الحجة عليهم بما وقفهم على خلق خلق .
The repetition of this verse is to emphasize (confirm) and exaggerate the confirmation and giving the proof on that he had inquired them about creating the Creation:وقال القتبي : إن الله تعالى عدد في هذه السورة نعماءه ، وذكر خلقه آلاءه ، ثم أتبع كل خلة وصفها ونعمة وضعها بهذه وجعلها فاصلة بين كل نعمتين لينبههم على النعم ويقررهم بها
Al-Qutaby said: in this surah Allah has enumerated his favours (or bounties) and informed about his creation of these favours and then he added to each manner HE has described and favour he has given this information (verse repetition) as a separator to remind them about the favours and settle them (Jinn and humans) on them (favours).كما تقول لمن تتابع فيه إحسانك وهو يكفره وينكره : ألم تكن فقيرا فأغنيتك ؛ أفتنكر هذا ؟ ! ألم تكن خاملا فعززتك ؛ أفتنكر هذا ؟ ! ألم تكن صرورة فحججت بك ؛ أفتنكر هذا ! ؟ ألم تكن راجلا فحملتك ؛ أفتنكر هذا ؟ ! والتكرير حسن في مثل هذا .
As if you said to a person you have done (or treated) well while he/she is denying it: Were you not poor and I made you rich; are you denying this? Have you not been motionless, but I made you stable? Are you denying this? Have you not failed to hajj, but I made you perform it in my company; Are you denying this? Have you not been a pedestrian, but I made you ride; Are you denying this? Repetition in these cases is good!
