I am confused with the grammar of the expression vielen Dank, because if Dank is a noun, and we have vielen, I would expect something like vielen Danken, or vielen Danke, or even vielen danke, for "many thanks" instead of "many thank".
Can you explain the logic or the rules for this?
The German word Dank is a masculine singular noun. It is not possible to use it in a plural form because there is no plural form. This is in heavy contrast to the English word thanks which is a plural noun of which no singular form exists. You can't use this English word as a singular because in English there is no singular of "thanks".
The German word viel is an undefined numeral, and if you use it in accusative case as attribute of a masculine noun, it becomes vielen.
But why is it in accusative case?
The phrase »vielen Dank« is an ellipsis. This means it is an incomplete sentence that can be completed to get a full sentence. Here it is:
Haben Sie vielen Dank! (Sie-Form)
Habe vielen Dank! (Du-Form)
verbatim: "Have many thanks!"
It is a sentence in imperative mode. You use imperative mode for commands and for wishes. In imperative mode the verb is not at position 2 but on position 1 and the subject ist almost always a pronoun in 2nd person that will be omitted if you use Du-Form. What is left is an object (»vielen Dank«) which is in accusative case because the verb haben needs it object to be in this case.
Addendum
In the comments the user dlatikay pointed out another fact that fits well to this question:
When ever you say "thank you" in English, you say "Danke" in German. Both phrases (the English and the German) are also ellipses. Here are the full sentences:
I thank you.
Ich danke dir. Ich danke Ihnen.
So here, the word »danke« is not a noun but a verb. It is a form of »danken« (Engl.: to thank (for something)). The reason, why you find it written with uppercase first letter very often is just, that the ellipsis, although it consists of only one word, is still a sentence, and the first letter of a sentence must be written with an uppercase first letter, like in English.
Btw.: The German word »Dankeschön« which is an extra polite and friendly way to say »thank you« (like in English: »thank you very much«) is a contraction of the verb »danke« as just discussed and the adjective »schön« which is used in the full sentence as an adverbial attribute of the verb. In this context you can translate it as »nicely«, »politely«, »friendly« but also »carefully«, »thoroughly«.
Dankeschön = Ich danke dir schön. Ich danke Ihnen schön.
I thank you nicely.
In modern German this meaning of »schön« slowly becomes extinct and is already very rare, but when I was a child (I was born 1965), I heard it often from older people like my grandparents, born between 1900 and 1910.
But »Dankeschön« also exists as a noun. When used as a noun, it means a gift or present that you give someone to express your gratitude.
That's simple: Dank has no plural in German (like a lot of similar nouns denoting a state of mind: Trotz, Gunst, Neid, Glück,... - and none of your examples for alternative expressions actually is a plural)
