Firstly, am I correct that xxte irun desu and xxte imasu are essentially identical in meaning, both 'polite', but the first possibly a bit more informal, colloquial & spoken as opposed to written?
But in 10.2 (& Ive seen the analogous construction elsewhere) Taro says おさがしですか? which seems to be the infinitive of sagasu followed by desu, apparently with exactly the same meaning as the other two constructions, 'looking for', continuing action.
Can someone explain a bit further? Are they identical, is there a 'nuance', or am I just wrong :-)
Infinitive+desu compared to ..te iru n desu or -te imasu, lesson 10.2

Hnrutt
July 20, 2012

Pascal-P
July 20, 2012
You can take the stem form of a verb and use it as a noun for a relative clause.
So "sagashi" means "search", as a noun. Therefore it means, "O (honorific) sagashi (search) desu ka (is it)?".
In the following audio lesson, I believe there's a similar case of "apaato sagashi wa dou?" to mean "How's the apartment search going?". Another case would be "omoshiroi hanashi" to mean "interesting talk/discussion".
So yeah, the infinitve form is kind of like a noun. I guess that makes sense since infinitives are independent of other parts of speech.
For your other question, "~te irun desu" and "~te imasu" are virtually the same.
One key point it'd like to make is that "te imasu" and "te irun desu" aren't unrelated. "te irun desu" is derived from "te imasu".
The formation of this expression is as follows:
Any Japanese verb in its dicitonary form can be followed by "no". The "no" also carries a sense of nominalization. Without getting too technical, the verb is made into a sort of pseudo-noun and then qualified with a "copula", as nouns tend to be if they're not paired with another verb.
Now because Japanese has this wonderful habit of shortening everything they say in everyday speaking, the "no desu" changes to "n desu", which also carries a more conversational, softer tone. The "no" and "n" carry the nuance of indicating an explanation, or a search for an explanation. (I'll explain in a moment).
Now, as you know, a continual action can be expressed by -te form + imasu. But "imasu" is just the polite form of "iru", "to be".
So then the "~te iru" verb can be combined with the explanatory "no" and the desu copula, and hence we have a "te irun desu" expression.
So an example which uses "no" and "n":
"Nande nemui no?" "Why are you sleepy" The "no" indicates asking for explanation, as opposed to "Nemui?" Which is just asking "Are you sleepy?" and thus isn't asking for explanation and hence doesn't carry the "no".
An answer could be "Takusan taberun deshita". "I ate a lot" The "n" here marks the explanation. The "deshita" indicates past tense, "ate".
As for the other variations of this:
"~te irun desu" is polite present.
"~te irun deshita" is polite past.
"~te irun da" is plain present.
"~te irun datta" is plain past.
So the tense of the construct is governed by the tense of "desu" copula. It can even be conditional plain and formal, "dattara/deshitara".

Hnrutt
July 23, 2012
Pascal, your replies are a gold mine! I wish I had some way to repay the time you spend on your excellent answers. (I do reply to queries in other fora & fields where I do have competence, so perhaps that's 'indirect' repayment, but not to you!)
I had spotted the iru(n)/imasu connection & got the general sense of the differences, but this is far more comprehensive.
The 'nuance' between imasu ka & iru n desu is clear above.
How does the infinitive desu ka version compare in 'nuance'? It *seems* very like the imasu ka form in the way its used.

Hnrutt
July 23, 2012
Ah, actually, Ive just figured there must still be something I dont get.
Your last example, "Takusan taberun deshita".
Surely that isnt any of these patterns is it? taberu is the dictionary form, not the infinitive (or -te form, but those wouldnt make sense in this reply) ....following the pattern described higher up, (sagashi desu) why isnt it
'Takusan tabe deshita' ? (Which somehow sounds odd, but the infinitive is tabe....)
I'm assuming one can put the sagashi desu form into the past tense.
Maybe usage differs between godan & ichidan verbs??
(I presumeably could also have just said 'takusan tabemashita'. Some nuance again probably :-) )

Pascal-P
July 23, 2012
Firstly, no problem. I enjoy helping people out. When I first started learning, I used to post on the French and Spanish forums, but now I've forgotten most of those languages =/. Also, when I first started, posting on forums helped me to learn more. Even if I didn't know the answer, I taught myself whatever the person was asking about, so that I could respond.
Secondly, I was really wondering whether I not I should have used "taberun deshita" as an example.
The reason I used it was because it's a case where the "~n/no" is necessary. It might not be one of the "~te imasu" or "~te irun desu" patterns, but think about how the "~te irun desu" pattern is just an extension of the "dictionary form + n/no desu", but with "iru".
This usage is not to be confused with "o sagashi desu" or "stem + desu". In this case, there's not really a nuance. Stem forms of verbs are used as nouns. Taro could have just as easily said "Apaato wo sagashite irun desu ka?". His deciding to use the stem form seems kind of arbitrary. I don't think there's any specific case where using the stem as a noun is required grammatically (since gerunds/nouns can be formed with "dictionary + koto", just as easily).
So basically, using "taberu" as an example,
"taberu (dictionary) no desu/deshita." or "taberun desu/deshita" "I eat/ate", implying an explanation for a completed action in the present or past.
"tabete iru no desu/deshita" or "tabeterun desu/deshita" "I am/was eating", also implying an explanation for an ongoing action in the present/past.
While "tabete imasu" just means "I'm eating".
I've never seen the stem form of "taberu" used specifically as a noun (probably because, as you say, it sounds awkward), so instead I'll use "hanasu".
"Hanashi desu" "It's a story".
So often using the stem as a noun has different connotations, to say, just the gerund of a verb.
"Sakana wa, doko ni mitsukatta no?" "Where did you find the fish?"
"Sono kaeri ni, ike de asatte irun deshita. Soshite, sakana ga gohiki wo te ni ireta"
"When returning/On the way back (lit. in that return), I was fishing in the pond (the ~te irun desu construction implies explanation), and I caught 5 fish! (literally, 5 fish entered my hand)".
"Sagashi desu" "It's a search". So Taro says "Is it a search?" "O sagashi desu ka?"
Beyond this, however, I'm not really sure what to say. I don't think there's a nuance to its usage, but I'm not too sure.

Hnrutt
July 24, 2012
Again, thanks.
Im trying to run before I can walk, but I always do that :-)
In MnN I just got to lesson 20 & the plain form, & it does complicate life horribly after sticking to teinego all this time. But on the other hand its quite clearly why I often cant understand what other people are saying to each other (in plain) as opposed to what they say to me (in teinego - & more slowly! When its not keigo/kenjougo; then Im in trouble again :-( )
For films & TV plain is essential it's clear.
I guess there are hardly any circumstances where I would actually *speak* in plain, as an 'older' & fairly 'senior' visitor. Although perhaps it is used speaking to small children (and maybe animals like tu in French; but I dont talk to Japanese animals a lot!)