"て" in どういたしまして


I have read on other threads here (as well as on other pages online) that the
て in expressions such as どういたしまし **て** , 初めまし **て** or あけまし **て** おめでとうございます
was not the て-form but rather "a final particle used for a returning remark"
(as stated in this thread <https://japanese.stackexchange.com/a/13055/54803>),
where the person who answered names the particle 反問的用法の終助詞.

Could you please say a bit more on that particle, its etymology, what it means
and how it is actually used?




て in どういたしまして are different from the other two.

[大辞泉](https://dictionary.goo.ne.jp/word/%E3%81%A6/#jn-148945) definition only
gives the meaning as a question marker: 質問や確かめの気持ちを表す。「あなたにもでき―」「いらしたことあっ―」
(FYI this is normally used by females).

反問的用法 should be a variant of this and interprets どういたしまして as 'What have I done
(to be thanked)?'. It is not possible to understand 初めまして or あけまして this way.

* * *

The other two are more simply te-form, indicating the sentence continues
(初めましてよろしくお願いします).

* * *

Etymology can only be guessed, but all of the above (終助詞 or not) seem to have
derived from renyo-kei (stem of te-form) of つ, auxiliary verb for perfective
(roughly modern た): [〘接助〙
(完了の助動詞「つ」の連用形から)](https://kotobank.jp/word/%E3%81%A6-573101).

Note all the expressions can be translated with perfect tense: What _have_ I
_done_?, Now we _have met_ for the first time, Now the new year _has come_.

