I hope it is ok to ask for a translation. there is a song named 打上花火 that I'm having problems to translate its title. my guess is launched firework? (打上げ + 花火) but where the げ went?
Generally, we don't answer "can you translate this" questions, for different reasons, unless the question itself shows that you have attempted to do the work yourself, but I wanted to address your question about the "disappearance" of げ.
うちあげ 打ち上げ 打上げ 打上 打ちあげ the word can be written in different ways. Typically, in Japanese, four-kanji combinations are preferred in many cases. There is a fondness for the clean elegance of "four kanji expressions". There is even a special category of idioms that are made of only four kanji, known as yojijukugo (Japanese: 四字熟語). Technically 打上花火 does not count as one of these, as it does not express a complete idiomatic thought, but the popularity of the form makes it an understandable choice for the title of a song.
As far as your translation goes, "launched firework" is fine... though I might try for a more poetic title as a matter of choice, something like "Fireworks in the Sky" or some such....
Additionally, since in English we don't generally distinguish between different types of fireworks or fireworks in different states, (for example, the cheap store bought hand-held fireworks are still just "fireworks", the giant shells that are prepared for large displays are still "fireworks", and when they explode they are still "fireworks"), it would not be strange for you to translate 打上花火 simply as "Fireworks" if you wished.
There are several 花火 products in Japan which are popular pastime playing at night in summer. They are classified in three categories as authentic classification.
- 打上花火 うちあげはなび
- 仕掛花火 しかけはなび
- 玩具花火 おもちゃはなび
However, 打上花火 is sometime considered as opposite concept to 手持花火 （てもちはなび）which means hand-held fireworks. I hope you could find definitions for each of above said words from reference sources.
For your translation work, probably you have to find specific definition of the target word by comparing opposite concepts as above.
Then, for your question about where げ has gone.
When they are written as verb 打ち上げる or 打上げる, they show げ as part of conjugation. Mixing ひらがな as 送り仮名 works as guide line for reader to show its correct way of reading and at same time, to interpret 漢字 correctly.
When two verbs 打つ＋上げる are joined, the first step is to write 打ち上げる and this way of expression shows that this word is comprised from two words.
Next step is to omit ち from 打ち so that the word is written as 打上げる. Now, impression of compound word has gone away and this word seems as one word. Consequently, reader/listener will receive this as one word verb.
As an extension from this principle, when 打上げる＋花火 are joined to make うちあげはなび, 送り仮名 for verb 打上げ will be omitted so that 打上花火 is formed as one word. Now, original character as verb that belongs to 打上げる has disappeared.
Please do not misunderstand that above mechanism is something you must follow. It is not law or rule that everyone must follow. As a proof, you can find 打上花火、打上げ花火、打ち上げ花火 all of them in internet. You are entirely free to choose whichever way of writing you want. Behind its difference, connotation of how strong the sense of each original verb is maintained.