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The question [～うございます - keigo
い-adjectives](https://japanese.stackexchange.com/questions/765) explains how
有難う is derived from 有難く as following:

  * ありがた **い**
  * ありがた **く** (連用形 of ありがたい forms an adverbial)
  * ありがた **う** (く changes to う)
  * ありが **と** う (う changes preceding kana to お-row kana

What is the history of this type of transformation?
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etymology morphology
Historical prospective of ウ音便 transformation (e.g. 有難く => 有難う)
This is a kind of sound shift known in Japanese as 音便{おんびん}. The [Japanese
Wikipedia
article](https://ja.wikipedia.org/wiki/%E9%9F%B3%E4%BE%BF#.E9.9F.B3.E5.BD.A2.E4.B8.8A.E3.81.AE.E5.88.86.E9.A1.9E.E3.81.A8.E7.94.A8.E4.BE.8B)
has some good information, if you can read Japanese.

Historically, all the _-i_ adjectives used to end in _-ki_ for the attributive
form (the form used when directly modifying a noun), and end in _-ku_ for the
adverbial form. "Good dog" would thus be よきいぬ, and "well done" would thus be
よくした. In the [Heian period](https://en.wikipedia.org/wiki/Heian_period),
adjectives underwent two kinds of 音便{おんびん}, so-called イ音便{おんびん} and ウ音便{おんびん},
where adjectives lost the //k// sound in the attributive and adverbial forms,
resulting in よいいぬ and ようした.

The イ音便{おんびん} has persisted in the modern language, whereas the ウ音便{おんびん}
reverted back in eastern Japanese from some time around the [Muromachi
period](https://en.wikipedia.org/wiki/Muromachi_period) (or, alternatively,
ウ音便{おんびん} never manifested in eastern Japan and that evidence only appears in
the textual record from around this time). I believe that Kansai-ben still
exhibits ウ音便{おんびん}, and ウ音便{おんびん} does still persist in nationwide-standard
普通語{ふつうご} in certain expressions, such as お早{はよ}う (equivalent to お + 早{はや}く:
_hayaku_ → _hayau_ → _hayō_ ) or おめでとう (equivalent to お + 愛{め}でたく: _medetaku_
→ _medetau_ → _medetō_ ).

This sound change is historical and regular, both the disappearance of the
//k// in adjective forms, and the shift from //au// to //oː//. The vowel shift
also occurred in verbs, particularly in the volitional or presumptive form.
This was originally _-amu_ for 四段{よだん}活用{かつよう} or "quadrigrade verbs",
composed of _-a_ as the 未然形{みぜんけい} or incomplete form of the verb stem + _-mu_
as the auxiliary verb ending expressing the presumptive or volitional. This
_-amu_ later lost its //m//, becoming _-au_ , and this then underwent the
regular vowel shift to _-ō_.
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This is a general rule, **and** a historical change. This question is
accounted by historical development and dialectal difference.

Let's see some Classical Japanese conjugations and their Modern reflections:

```

    Forms     Classical       Modern Eastern      Modern Western
    終止形    たのし                (replaced by 連体形)
    連体形    たのしき        たのしい            たのしい
    連用形    たのしく        たのしく            たのしゅう
    
```

Classical adjectives had three distinct forms: 終止形 ("I **am happy** "), 連体形 ("
**happy** man") and 連用形 ("play **happily** "), among which 終止形 was eventually
replaced by 連体形, resulting in today's dictionary form that serves for both
functions.

Meanwhile, there was a sound change that eliminated consonant //k// and //g//
in the middle of word ([lenition](https://en.wikipedia.org/wiki/Lenition)). In
Western dialects (e.g. Kyoto or Osaka) this effect was active when the
following consonant is //-i// and //-u//, while in Eastern dialects (e.g.
Tokyo) it only happened when //-i//. As the result, たのしき became たのしい nation-
wide, but たのしく only became たのしう in Western area, remaining unchanged in
Eastern.

Next, there was another change that made diphthongs ended in //-u// to long
vowels. Thus in **Western dialects** , verbs' 連用形 (i.e. adverbial form) have
become:

  * ありがたう _arigat **au**_ → ありがとう _arigat **ō**_
  * たのしう _tanos **iu**_ → たのしゅう _tanos **yū**_
  * おもしろう _omosir **ou**_ → おもしろう _omosir **ō**_

So, in Modern context, ありがたく and ありがとう are Eastern and Western form of the
same conjugation. As Modern Standard Japanese is based on a Eastern dialect in
Tokyo, we only need ありがたく in theory. Why the form ありがとう appears in Standard
Japanese?

That's because some courtly language was imported from Kyoto, where the palace
was located until Meiji. Forms end with -う are basically only occur in very
limited occasion, namely before ございます, which is the highest variation of
keigo. The common greeting words like ありがとう or おはよう are originally followed by
ございます too, but come to be used independently for brevity.

* * *

[The dialect atlas by NINJAL, No. 139](http://www2.ninjal.ac.jp/hogen/dp/gaj-
pdf/gaj-map-legend/vol3/GAJ3-139.pdf). Distribution of adverbial form of an
adjective. Generally, blue-green area retains the consonant //k//, orange area
without //k//.

[![方言地図第139図](https://i.stack.imgur.com/YTgtA.png)](https://i.stack.imgur.com/YTgtA.png)
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